%------------------------------------------------------------------------------
% File     : ITP244_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Succ 00501_031916
% 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    : 0070_VEBT_Succ_00501_031916 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 13472 (4704 unt;2567 typ;   0 def)
%            Number of atoms       : 25554 (9996 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 15987 (2142   ~; 456   |;1267   &)
%                                         (2790 <=>;9332  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of FOOLs       : 1211 ( 587 fml; 624 var)
%            Number of X terms     :  782 (   0  []; 667 ite; 115 let)
%            Number of types       :  489 ( 488 usr)
%            Number of type conns  : 2402 (1636   >; 766   *;   0   +;   0  <<)
%            Number of predicates  :  123 ( 120 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1975 (1975 usr; 459 con; 0-8 aty)
%            Number of variables   : 26216 (25487   !; 729   ?;26216   :)
% 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
%            2022-02-18 00:38:22.816
%------------------------------------------------------------------------------
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tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(sy_c_ATP_058Lamm__aaj____,type,
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tff(sy_c_ATP_058Lamm__aba____,type,
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tff(sy_c_ATP_058Lamm__abb____,type,
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tff(sy_c_ATP_058Lamm__abc____,type,
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tff(sy_c_ATP_058Lamm__abd____,type,
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tff(sy_c_ATP_058Lamm__abf____,type,
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tff(sy_c_ATP_058Lamm__abg____,type,
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tff(sy_c_ATP_058Lamm__abh____,type,
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tff(sy_c_ATP_058Lamm__abj____,type,
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tff(sy_c_ATP_058Lamm__abo____,type,
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tff(sy_c_ATP_058Lamm__abp____,type,
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tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: fun_real_o ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: fun_real_o ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_real_real ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_real_real ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_nat_real ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: fun_nat_real ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: fun_nat_real ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_real_real ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_int_int ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: list_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: list_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: list_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: list_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: list_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: list_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: list_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: list_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_Pr1261947904930325089at_nat > fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_real_real ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_int_int ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_real_o > fun_fu6074783352538106518real_o ).

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

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_real_o > fun_fu4600449761433299222_nat_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_nat_fun_real_o > fun_real_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_real_o > fun_fu8967550341600084630plex_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_co4998825143550818160real_o > fun_re7319081385389557872plex_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_o > fun_fu4383628726152020410real_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_real_fun_nat_o > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_o > fun_fu1148935994063782970_nat_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_o > fun_fu2079201134200759482plex_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_co5676208753217392844_nat_o > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_complex_o > fun_fu4927436880037274520real_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_re7319081385389557872plex_o > fun_co4998825143550818160real_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_complex_o > fun_fu7221214265560853784_nat_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_na1554462459066888140plex_o > fun_co5676208753217392844_nat_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_complex_o > fun_fu8407643262223357336plex_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_co1148497455491383662plex_o > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_real_o > fun_fu5499870864232458646_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_li7444029373303974462real_o > fun_re4304251901236488766_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_real_o > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_real_o > fun_fu2498659858537143873_nat_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_real_o > fun_fu4846543092220166533plex_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_nat_o > fun_fu1031053159183118053real_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_nat_o > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_o > fun_fu80362980471826017plex_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_complex_o > fun_fu9102338020654974087real_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_complex_o > fun_fu2734159213235317439_nat_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_complex_o > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_real_o > fun_fu1123161595235077537_nat_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_real_o > fun_fu8020951434679919326real_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_real_o > fun_fu3654966910337544342_nat_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_real_o > fun_fu8168279061277983450plex_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_real_o > fun_fu7307196158820694110real_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_real_o > fun_fu2993764712125629974_nat_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_real_o > fun_fu8433555683852769370plex_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_real_o > fun_fu8632917424891031774real_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_real_o > fun_fu226109141602483862_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_real_o > fun_fu2587577306029068506plex_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_nat_o > fun_fu4877945105896157498real_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_nat_o > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_int > fun_fu4228744305898656589_int_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_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_int > fun_fu7714913403879666293_int_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_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_nat_real > fun_fu3306995184869731826t_real ).

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

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_real_complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_real_rat > fun_real_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_real_int > fun_real_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_complex_int > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_nat_complex > fun_nat_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_int_int > fun_int_int ).

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

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

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

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

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_nat > fun_fu1217771548038759377at_nat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_na3970208099681795815_nat_o > fun_fu484727072080742171_nat_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_fu1115759055147092071_o_nat > fun_fu4228150600790569371at_nat ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_na8713714802964096592at_nat > fun_fu1039311746440044420at_nat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_fu1413608398870722896at_nat > fun_fu7448223259553364356at_nat ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_Pr2602117396471556196at_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_nat > fun_fu7002938529518989751at_nat ).

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

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

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

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: complex > fun_na1142814060743658952omplex ).

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

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

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_real_rat > fun_fu2598743018846156927al_rat ).

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

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

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_int > fun_nat_int ).

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

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

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

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_real > fun_nat_real ).

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

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

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

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: complex > fun_co3258106519541737638omplex ).

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

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: nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: rat > fun_nat_fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_nat_real > fun_na5225841814499866095t_real ).

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

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: nat > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_int_int ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_real_real ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: rat > fun_nat_fun_rat_rat ).

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

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

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: rat > fun_nat_fun_rat_rat ).

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

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_int_int ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_int > fun_int_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_rat > fun_nat_fun_rat_rat ).

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

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_complex_complex > fun_fu3719809307426920623omplex ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_real > fun_fu4002475321494201261t_real ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: complex > fun_fu4781654129426620053omplex ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: int > fun_fu6415725884910174923nt_int ).

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

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_nat_rat > fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_real_complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_complex_complex > fun_complex_o ).

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

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_real_rat > fun_real_o ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_real_int > fun_real_o ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_complex_int > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_Pr672294278902479040t_real > fun_fu4217534345154246979t_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_int_complex > fun_fu3594690983434520489t_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_real_complex > fun_fu6344110373156359337l_real ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_complex_complex > fun_fu8968174646369908141x_real ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_Pr4050773816208733250omplex > fun_fu7180905015247952065t_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_nat_complex > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_nat_complex > fun_na1554462459066888140plex_o ).

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

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: rat > fun_nat_rat ).

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: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_int > fun_int_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: nat > fun_nat_fun_nat_nat ).

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

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

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: nat > fun_fu3593947265352647275omplex ).

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_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: nat > fun_fu6810213084838398785t_real ).

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

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_real_o ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: code_integer > fun_o_7148296245751160776un_o_o ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: num > fun_nu7188104440484974293_num_o ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: nat > fun_nu2628545618185965215_num_o ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_int_nat > fun_fu9189860104212011539nt_nat ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_real_nat > fun_fu6296426149269091475al_nat ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_complex_nat > fun_fu2872759208803094421ex_nat ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_int_nat > fun_fu895286674026184139t_real ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_real_nat > fun_fu3338195423232944715l_real ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_complex_nat > fun_fu2443423965090536013x_real ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_int_real > fun_fu923666292223928999nt_int ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: fun_int_complex > fun_fu424603779095247273nt_int ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_int_nat > fun_fu1427849550841218251nt_int ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_int_int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_in1696471567231710611nt_int ).

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

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

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

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: 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: rat > int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: complex > fun_nat_complex ).

tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > fun_nat_int ).

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > fun_nat_rat ).

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

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

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

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: fun_num_fun_num_num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > fun_int_int ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: int > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: int > fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
    bit_un1837492267222099188nd_num: fun_nu1049378303802735384on_num ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
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    bit_un2480387367778600638or_num: fun_nu1049378303802735384on_num ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

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

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

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

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

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

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

tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
    comple4887499456419720421f_real: set_real > real ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

tff(sy_c_Complex_OArg,type,
    arg: fun_complex_real ).

tff(sy_c_Complex_Ocis,type,
    cis: fun_real_complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: fun_complex_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: fun_re7812779276277946752omplex ).

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: fun_complex_real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: fun_complex_real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: fun_complex_complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

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

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

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: int > fun_int_int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
    unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: fun_Pr3238374608701176162_int_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: ( num * num ) > produc8923325533196201883nteger ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: ( num * num ) > product_prod_int_int ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: ( num * num ) > product_prod_nat_nat ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: ( num * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: num > fun_Pr2478283820680522243nt_int ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
    unique5026877609467782581ep_nat: ( num * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
    comm_s8582702949713902594nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > fun_nat_complex ).

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tff(sy_c_List_Olist__update_001_Eo,type,
    list_update_o: ( list_o * nat * $o ) > list_o ).

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

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

tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: ( list_nat * nat * nat ) > list_nat ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: ( list_real * nat * real ) > list_real ).

tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
    list_u1324408373059187874T_VEBT: ( list_VEBT_VEBT * nat * vEBT_VEBT ) > list_VEBT_VEBT ).

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__Complex__Ocomplex,type,
    nth_complex: list_complex > fun_nat_complex ).

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

tff(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: ( list_list_nat * nat ) > list_nat ).

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_I_Eo_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_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__Num__Onum_Mt__Num__Onum_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,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > fun_nat_real ).

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

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

tff(sy_c_List_Oproduct_001_Eo_001_Eo,type,
    product_o_o: ( list_o * list_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
    product_o_nat: ( list_o * list_nat ) > list_P6285523579766656935_o_nat ).

tff(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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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__Nat__Onat_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
    product_num_num: ( list_num * list_num ) > list_P3744719386663036955um_num ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_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_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

tff(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: ( nat * $o ) > list_o ).

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

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

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

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_o > fun_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_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_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

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

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    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_Olist__encode,type,
    nat_list_encode: fun_list_nat_nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun_li1025265955262268170_nat_o ).

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

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

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

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

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

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

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

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

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

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

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

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

tff(sy_c_Num_Opow,type,
    pow: fun_num_fun_num_num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

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

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

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

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: ( $o * fun_Pr4796943543331071514_nat_o ) > fun_op8028897214545316308_nat_o ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_M_Eo_J_J,type,
    bot_bo897209107778963633un_o_o: fun_Co480733329897759773un_o_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    bot_bo428638681919004382_int_o: fun_int_fun_int_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    bot_bo1118997454112379798_nat_o: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    bot_bo1877178017081890764_num_o: fun_nat_fun_num_o ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    bot_bo3657453288633029378_num_o: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > fun_Extended_enat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: ( set_Code_integer * set_Code_integer ) > $o ).

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

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

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le7866589430770878221at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > $o ).

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > fun_Extended_enat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: ( set_o * set_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: ( set_Code_integer * set_Code_integer ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: ( set_complex * set_complex ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: ( set_int * set_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    ord_le8980329558974975238eger_o: ( set_Pr448751882837621926eger_o * set_Pr448751882837621926eger_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: ( set_Pr958786334691620121nt_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    ord_le8085105155179020875at_num: ( set_Pr6200539531224447659at_num * set_Pr6200539531224447659at_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
    ord_le880128212290418581um_num: ( set_Pr8218934625190621173um_num * set_Pr8218934625190621173um_num ) > $o ).

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

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

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

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_M_Eo_J_J,type,
    ord_le2909477399203599485un_o_o: ( fun_Co480733329897759773un_o_o * fun_Co480733329897759773un_o_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    ord_le77980682512674322_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    ord_le768339454706049738_nat_o: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    ord_le1526520017675560704_num_o: ( fun_nat_fun_num_o * fun_nat_fun_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    ord_le3306795289226699318_num_o: ( fun_num_fun_num_o * fun_num_fun_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: fun_Ex8341929632020906428d_enat ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: fun_num_fun_num_num ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

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

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

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

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

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: fun_in1696471567231710611nt_int ).

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

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

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

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

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

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

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc7967433646104525454on_nat: ( fun_nat_fun_nat_o * produc4953844613479565601on_nat ) > produc1615844007061411358on_nat ).

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

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc1140783415258738358on_num: ( fun_num_fun_num_o * produc3447558737645232053on_num ) > produc50614890452615238on_num ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc6360833703471310868on_num: ( fun_num_fun_num_num * produc3447558737645232053on_num ) > produc2703417854250991842on_num ).

tff(sy_c_Product__Type_OPair_001t__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_M_Eo_J_J_001t__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,type,
    produc3653971535690468304at_nat: ( fun_Pr2028404973800837934_nat_o * produc6121120109295599847at_nat ) > produc3739950874144486240at_nat ).

tff(sy_c_Product__Type_OPair_001t__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_001t__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,type,
    produc7814092408141220007at_nat: ( fun_Pr2477239776880934807at_nat * produc6121120109295599847at_nat ) > produc1260151622200656695at_nat ).

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

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc1043322548047392435omplex: ( fun_Co5015990272074005067omplex * produc6271795597528267376eger_o ) > set_complex ).

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

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

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( fun_Co3544761417380347684eger_o * produc8923325533196201883nteger ) > produc6271795597528267376eger_o ).

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: fun_nat_fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: fun_nat_fun_nat_int > fun_Pr7647638413817135296at_int ).

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

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

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

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4528391901023682385at_nat: fun_na1668100129766719005at_nat > fun_Pr2477239776880934807at_nat ).

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
    produc9175805072616146554et_int: ( fun_na8306170748371236422et_int * product_prod_nat_num ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4130284055270567454et_nat: ( fun_na3896347130304379370et_nat * product_prod_nat_num ) > set_nat ).

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

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

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

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

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun_Pr1652463973008025892nt_rat ).

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

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

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_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: rat > real ).

tff(sy_c_Rat_Onormalize,type,
    normalize: fun_Pr2478283820680522243nt_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

tff(sy_c_Rat_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: product_prod_int_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

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

tff(sy_c_Real__Vector__Spaces_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_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: fun_re7208826350674946562omplex ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: fun_re7681423839302404990l_real ).

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: ( code_integer * code_integer ) > code_integer ).

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

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: 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 * 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,
    dvd_dvd_complex: fun_co1148497455491383662plex_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: int > fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Set_OCollect_001_Eo,type,
    collect_o: fun_o_o > set_o ).

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

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

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

tff(sy_c_Set_OCollect_001t__List__Olist_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,
    collec5608196760682091941T_VEBT: fun_list_VEBT_VEBT_o > set_list_VEBT_VEBT ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_o > set_nat ).

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

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

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

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

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

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

tff(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
    collect_VEBT_VEBT: fun_VEBT_VEBT_o > set_VEBT_VEBT ).

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__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    image_list_nat_nat: ( fun_list_nat_nat * set_list_nat ) > set_nat ).

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

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

tff(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_2486076414777270412at_nat: ( fun_Pr2602117396471556196at_nat * set_Pr1261947904930325089at_nat ) > set_nat ).

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

tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

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

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

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

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

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

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

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

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

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

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

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

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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

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

tff(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
    topolo4899668324122417113eq_int: fun_nat_int > $o ).

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

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

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

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

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

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

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

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

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

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

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

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

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

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

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

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__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_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__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option_nat ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

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

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr9018117161326028078_int_o > fun_Pr3238374608701176162_int_o ).

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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
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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 ).

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tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Num__Onum,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Rat__Orat,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Complex__Ocomplex_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun6946558556880649704l_real: ( fun_fu3077872570260440615l_real * fun_real_real ) > fun_real_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_M_Eo_J_J,type,
    aa_fun3693225987673163702real_o: ( fun_fu5533088652784741581real_o * fun_real_real ) > fun_fu3022329567491847689real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_M_Eo_J_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun7329550764591405201real_o: ( fun_fu4927436880037274520real_o * fun_re7319081385389557872plex_o ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    aa_fun434462017428657539plex_o: ( fun_fu5980176762240715674plex_o * fun_re7208826350674946562omplex ) > fun_complex_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun6402932185680802879_nat_o: ( fun_fu3188172205177979734_nat_o * fun_re175488254389786404ex_nat ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun2419829551541144455real_o: ( fun_fu8962730947843575454real_o * fun_re2354894731504631552x_real ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun8982401453414893235real_o: ( fun_fu4383628726152020410real_o * fun_real_fun_nat_o ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    aa_fun2810168320254036705plex_o: ( fun_fu9045617033751531768plex_o * fun_re2084383175013312036omplex ) > fun_complex_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun1675651493036827037_nat_o: ( fun_fu3065505572990288052_nat_o * fun_real_fun_nat_nat ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun3910182461874428645real_o: ( fun_fu2658902142918880764real_o * fun_re4238438383274954274t_real ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Real__Oreal_M_Eo_J_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun8588574755773516687real_o: ( fun_fu6074783352538106518real_o * fun_real_fun_real_o ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J_J_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    aa_fun890640625120065285plex_o: ( fun_fu6215981411654662428plex_o * fun_re7812779276277946752omplex ) > fun_complex_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Real__Oreal_Mt__Nat__Onat_J_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun6822315102276328897_nat_o: ( fun_fu1396458009202489560_nat_o * fun_re7428432522468557474al_nat ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun6833002213068288777real_o: ( fun_fu8737060993192706592real_o * fun_re7681423839302404990l_real ) > fun_real_o ).

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

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

tff(sy_c_member_001t__Int__Oint,type,
    member_int: ( int * set_int ) > $o ).

tff(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: ( list_o * set_list_o ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: ( list_int * set_list_int ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_o ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    member1379723562493234055eger_o: ( produc6271795597528267376eger_o * set_Pr448751882837621926eger_o ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    member9148766508732265716at_num: ( product_prod_nat_num * set_Pr6200539531224447659at_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: ( product_prod_num_num * set_Pr8218934625190621173um_num ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

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

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: ( set_nat * set_set_nat ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_sxa____,type,
    sxa: nat ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10904)
tff(fact_0__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = na ).

% \<open>deg div 2 = n\<close>
tff(fact_1__C5_Ohyps_C_I3_J,axiom,
    ! [Xb: nat,Sxb: nat] :
      ( ( vEBT_vebt_succ(summary,Xb) = some_nat(Sxb) )
    <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(summary)),Xb),Sxb) ) ).

% "5.hyps"(3)
tff(fact_2__C5_Ohyps_C_I9_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5.hyps"(9)
tff(fact_3_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_4_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_5_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( some_nat(X2) = some_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_6_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_7_prod_Oinject,axiom,
    ! [X1: code_integer,X2: $o,Y1: code_integer,Y2: $o] :
      ( ( produc6677183202524767010eger_o(X1,(X2)) = produc6677183202524767010eger_o(Y1,(Y2)) )
    <=> ( ( X1 = Y1 )
        & ( (X2)
        <=> (Y2) ) ) ) ).

% prod.inject
tff(fact_8_prod_Oinject,axiom,
    ! [X1: num,X2: num,Y1: num,Y2: num] :
      ( ( product_Pair_num_num(X1,X2) = product_Pair_num_num(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_9_prod_Oinject,axiom,
    ! [X1: nat,X2: num,Y1: nat,Y2: num] :
      ( ( product_Pair_nat_num(X1,X2) = product_Pair_nat_num(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_10_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_11_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_12_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
      ( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A2,(B2)) )
    <=> ( ( A = A2 )
        & ( (B)
        <=> (B2) ) ) ) ).

% old.prod.inject
tff(fact_13_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_14_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_15_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_16_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_17__C5_Ohyps_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(vEBT_invar_vebt(X),na)
        & ! [Xa: nat,Xb2: nat] :
            ( ( vEBT_vebt_succ(X,Xa) = some_nat(Xb2) )
          <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(X)),Xa),Xb2) ) ) ) ).

% "5.hyps"(1)
tff(fact_18_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Maa) ).

% vebt_maxt.simps(3)
tff(fact_19__C5_Ohyps_C_I2_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

% "5.hyps"(2)
tff(fact_20_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Mia) ).

% vebt_mint.simps(3)
tff(fact_21_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] : Xb != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),M) ).

% prod_decode_aux.cases
tff(fact_22__C5_Ohyps_C_I6_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "5.hyps"(6)
tff(fact_23__C5_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ).

% "5.hyps"(8)
tff(fact_24_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6271795597528267376eger_o] :
      ~ ! [A3: code_integer,B3: $o] : Y != produc6677183202524767010eger_o(A3,(B3)) ).

% old.prod.exhaust
tff(fact_25_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_num_num] :
      ~ ! [A3: num,B3: num] : Y != product_Pair_num_num(A3,B3) ).

% old.prod.exhaust
tff(fact_26_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_num] :
      ~ ! [A3: nat,B3: num] : Y != product_Pair_nat_num(A3,B3) ).

% old.prod.exhaust
tff(fact_27_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) ).

% old.prod.exhaust
tff(fact_28_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) ).

% old.prod.exhaust
tff(fact_29_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_30_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)
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),X3) ) ) ) ).

% min_in_set_def
tff(fact_31_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)
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xb) ) ) ) ).

% max_in_set_def
tff(fact_32_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb) ) ).

% maxbmo
tff(fact_33_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_34__092_060open_0622_A_092_060le_062_Adeg_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))),deg) ).

% \<open>2 \<le> deg\<close>
tff(fact_35_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) )
          & ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_36_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( 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_37_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( 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_38_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( 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_39_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( 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_40__C5_Ohyps_C_I5_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "5.hyps"(5)
tff(fact_41_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_42_Pair__inject,axiom,
    ! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
      ( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A2,(B2)) )
     => ~ ( ( A = A2 )
         => ( (B)
          <=> ~ (B2) ) ) ) ).

% Pair_inject
tff(fact_43_Pair__inject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_44_Pair__inject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_45_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_46_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_47_prod__cases,axiom,
    ! [P: fun_Pr3290071492984375169er_o_o,P2: produc6271795597528267376eger_o] :
      ( ! [A3: code_integer,B3: $o] : aa_Pro5286387174248033258er_o_o(P,produc6677183202524767010eger_o(A3,(B3)))
     => aa_Pro5286387174248033258er_o_o(P,P2) ) ).

% prod_cases
tff(fact_48_prod__cases,axiom,
    ! [P: fun_Pr6331313853612757382_num_o,P2: product_prod_num_num] :
      ( ! [A3: num,B3: num] : aa_Pro7685007914727650815_num_o(P,product_Pair_num_num(A3,B3))
     => aa_Pro7685007914727650815_num_o(P,P2) ) ).

% prod_cases
tff(fact_49_prod__cases,axiom,
    ! [P: fun_Pr5555124106300582480_num_o,P2: product_prod_nat_num] :
      ( ! [A3: nat,B3: num] : aa_Pro6908818167415475913_num_o(P,product_Pair_nat_num(A3,B3))
     => aa_Pro6908818167415475913_num_o(P,P2) ) ).

% prod_cases
tff(fact_50_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_51_prod__cases,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A3: int,B3: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3))
     => aa_Pro4592068669816069595_int_o(P,P2) ) ).

% prod_cases
tff(fact_52_surj__pair,axiom,
    ! [P2: produc6271795597528267376eger_o] :
    ? [X4: code_integer,Y3: $o] : P2 = produc6677183202524767010eger_o(X4,(Y3)) ).

% surj_pair
tff(fact_53_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X4: num,Y3: num] : P2 = product_Pair_num_num(X4,Y3) ).

% surj_pair
tff(fact_54_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X4: nat,Y3: num] : P2 = product_Pair_nat_num(X4,Y3) ).

% surj_pair
tff(fact_55_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y3: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) ).

% surj_pair
tff(fact_56_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X4: int,Y3: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) ).

% surj_pair
tff(fact_57_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_58_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_59_add__self__div__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).

% add_self_div_2
tff(fact_60_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( member_int(A,collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_61_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( member8440522571783428010at_nat(A,collec3392354462482085612at_nat(P))
    <=> aa_Pro6150637604445964947_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_62_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( member_complex(A,collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_63_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( member_real(A,collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_64_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> aa_list_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_65_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_66_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_67_Collect__mem__eq,axiom,
    ! [A4: set_int] : collect_int(aTP_Lamm_a(A4)) = A4 ).

% Collect_mem_eq
tff(fact_68_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_aa(A4)) = A4 ).

% Collect_mem_eq
tff(fact_69_Collect__mem__eq,axiom,
    ! [A4: set_complex] : collect_complex(aTP_Lamm_ab(A4)) = A4 ).

% Collect_mem_eq
tff(fact_70_Collect__mem__eq,axiom,
    ! [A4: set_real] : collect_real(aTP_Lamm_ac(A4)) = A4 ).

% Collect_mem_eq
tff(fact_71_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] : collect_list_nat(aTP_Lamm_ad(A4)) = A4 ).

% Collect_mem_eq
tff(fact_72_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] : collect_set_nat(aTP_Lamm_ae(A4)) = A4 ).

% Collect_mem_eq
tff(fact_73_Collect__mem__eq,axiom,
    ! [A4: set_nat] : collect_nat(aTP_Lamm_af(A4)) = A4 ).

% Collect_mem_eq
tff(fact_74_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X4: complex] :
          ( aa_complex_o(P,X4)
        <=> aa_complex_o(Q,X4) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_75_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X4: real] :
          ( aa_real_o(P,X4)
        <=> aa_real_o(Q,X4) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_76_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
        <=> aa_list_nat_o(Q,X4) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_77_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
        <=> aa_set_nat_o(Q,X4) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_78_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(P,X4)
        <=> aa_nat_o(Q,X4) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_79_succ__min,axiom,
    ! [Dega: nat,Xb: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Mia) ) ) ) ).

% succ_min
tff(fact_80_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( 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_81_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( 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_82_nat__add__left__cancel__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_83_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_84_field__sum__of__halves,axiom,
    ! [Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(Xb,numeral_numeral_rat(aa_num_num(bit0,one)))),divide_divide_rat(Xb,numeral_numeral_rat(aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_85_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa_num_num(bit0,Mb) != one ).

% semiring_norm(85)
tff(fact_86_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_87_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_88_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_89_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Mb)),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_90_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_91_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_92_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_93_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_94_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_95_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_96_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_97_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_98_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_99_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_100_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_101_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_102_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Mb) = numera6690914467698888265omplex(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_103_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numeral_numeral_real(Mb) = numeral_numeral_real(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_104_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numeral_numeral_rat(Mb) = numeral_numeral_rat(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_105_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Mb) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_106_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Mb) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_107_semiring__norm_I87_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit0,Mb) = aa_num_num(bit0,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(87)
tff(fact_108_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa_nat_nat(suc,Nat) = aa_nat_nat(suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_109_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_110_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_111_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_112_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_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,Xb),Y)
        & ! [Z2: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z2) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z2) ) ) ) ).

% succ_member
tff(fact_113_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_114_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Mb)),numeral_numeral_rat(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_115_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_116_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_117_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_118_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Mb)),numeral_numeral_rat(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_119_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_120_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_121_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_122_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_123_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_124_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_125_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_126_nat__add__left__cancel__less,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_127_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(6)
tff(fact_128_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_129_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% Suc_numeral
tff(fact_130_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_131_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_132_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_133_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_134_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_135__C5_Ohyps_C_I10_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.hyps"(10)
tff(fact_136_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_137_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_138_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q2: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ) ).

% less_natE
tff(fact_139_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_140_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_141_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_142_less__add__Suc1,axiom,
    ! [I: nat,Mb: 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),Mb))) ).

% less_add_Suc1
tff(fact_143_less__add__Suc2,axiom,
    ! [I: nat,Mb: 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,Mb),I))) ).

% less_add_Suc2
tff(fact_144_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_145_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_146_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_147_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_148_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_149_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)) ) ).

% less_imp_Suc_add
tff(fact_150_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_151_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_152_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_153_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: 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,Nb) ) ).

% infinite_descent
tff(fact_154_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: 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,Nb) ) ).

% nat_less_induct
tff(fact_155_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_156_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,K: 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),K)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_157_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K2) ) ) ).

% less_trans_Suc
tff(fact_158_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_159_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_160_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_161_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_162_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_163_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M3: nat] :
          ( ( Mb = aa_nat_nat(suc,M3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M3) ) ) ).

% Suc_less_eq2
tff(fact_164_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
           => aa_nat_o(P,I3) ) ) ) ).

% All_less_Suc
tff(fact_165_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_166_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_167_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_168_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_169_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
            & aa_nat_o(P,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_170_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_171_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_172_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_173_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_174_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K2)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K2 != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_175_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_176_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
     => ( ( K2 != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K2 != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_177_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_178_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_179_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_180_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_181_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_182_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: 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,Nb),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_183_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: 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,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_184_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: 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,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_185_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: 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,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_186_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: 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,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_187_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one) ).

% add_One_commute
tff(fact_188_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X4),X4)
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X4),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X4),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,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_189_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P,Mb)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_190_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: 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,Nb) ) ).

% full_nat_induct
tff(fact_191_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_192_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_193_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_194_Suc__le__D,axiom,
    ! [Nb: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M4)
     => ? [M: nat] : M4 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_195_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_196_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_197_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_198_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_199_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_200_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(suc,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_201_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_202_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_203_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_204_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_205_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_206_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_207_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_208_trans__less__add2,axiom,
    ! [I: nat,J: nat,Mb: 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,Mb),J)) ) ).

% trans_less_add2
tff(fact_209_trans__less__add1,axiom,
    ! [I: nat,J: nat,Mb: 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),Mb)) ) ).

% trans_less_add1
tff(fact_210_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_less_mono1
tff(fact_211_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_212_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_213_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_214_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: 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)),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2) ) ).

% add_lessD1
tff(fact_215_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_216_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_217_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: 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,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_218_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: 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,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_219_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: 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,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_220_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: 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,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_221_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N2: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Nb)),aa_nat_set_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_222_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: 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,Nb),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_223_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: 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,Nb),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_224_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: 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,Nb),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_225_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: 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,Nb),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_226_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K2: nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Mb)),K2)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2))) ) ).

% mono_nat_linear_lb
tff(fact_227_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_228_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_229_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% is_num_normalize(1)
tff(fact_230_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% is_num_normalize(1)
tff(fact_231_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_232_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => ? [X4: nat] :
            ( aa_nat_o(P,X4)
            & ! [Y4: nat] :
                ( aa_nat_o(P,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_233_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_234_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_235_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_236_le__trans,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2) ) ) ).

% le_trans
tff(fact_237_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_238_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)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_239_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),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_240_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3) ) ).

% nat_le_iff_add
tff(fact_241_trans__le__add2,axiom,
    ! [I: nat,J: nat,Mb: 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,Mb),J)) ) ).

% trans_le_add2
tff(fact_242_trans__le__add1,axiom,
    ! [I: nat,J: nat,Mb: 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),Mb)) ) ).

% trans_le_add1
tff(fact_243_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_le_mono1
tff(fact_244_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: 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,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_245_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N) ) ).

% le_Suc_ex
tff(fact_246_add__leD2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% add_leD2
tff(fact_247_add__leD1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_248_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_249_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_250_add__leE,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% add_leE
tff(fact_251_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_252_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K2)) ) ).

% div_le_mono
tff(fact_253_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_254_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_real(aa_num_num(bit0,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),numeral_numeral_real(Nb)) ).

% numeral_Bit0
tff(fact_255_numeral__Bit0,axiom,
    ! [Nb: num] : numeral_numeral_rat(aa_num_num(bit0,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Nb)),numeral_numeral_rat(Nb)) ).

% numeral_Bit0
tff(fact_256_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_257_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_258_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_259_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(one)) = A ).

% divide_numeral_1
tff(fact_260_divide__numeral__1,axiom,
    ! [A: rat] : divide_divide_rat(A,numeral_numeral_rat(one)) = A ).

% divide_numeral_1
tff(fact_261_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_262_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_263_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_264_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_265_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_266_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( some_nat(Mb) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% misiz
tff(fact_267_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_268_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_269_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_270_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_271_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_272_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_273_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_274_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_275_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_276_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_277_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_278_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_279_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_280_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_281_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_282_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_283_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_284_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_285_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_286__C5_Ohyps_C_I7_J,axiom,
    ! [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))),m))
     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),X_12)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I4) ) ) ).

% "5.hyps"(7)
tff(fact_287_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_288_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_289_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_290_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_291_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_292_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_293_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_294_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_295_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_296_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_297_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_298_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_299_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_300_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),one) ).

% semiring_norm(75)
tff(fact_301_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_302_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_303_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).

% semiring_norm(69)
tff(fact_304__C5_Ohyps_C_I4_J,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "5.hyps"(4)
tff(fact_305_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_306_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_307_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_308_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_309_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_311_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_312_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_313_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_314_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_315_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_316_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_317_group__cancel_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_318_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A4),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_319_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_320_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_321_group__cancel_Oadd2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_322_group__cancel_Oadd2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_323_group__cancel_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_324_group__cancel_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_325_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.assoc
tff(fact_326_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_327_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_328_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_329_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_330_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_331_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_332_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_333_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_334_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_335_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).

% add.commute
tff(fact_336_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% add.commute
tff(fact_337_add_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% add.commute
tff(fact_338_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% add.commute
tff(fact_339_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% add.left_commute
tff(fact_340_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.left_commute
tff(fact_341_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_342_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_343_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_344_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_345_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_346_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_347_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_348_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_349_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_350_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_351_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_352_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_353_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K2 = 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_354_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K2 = 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),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_355_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_356_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_357_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_358_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),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),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_359_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_360_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_361_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_362_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K2: 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,K2),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),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_363_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_364_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_mono
tff(fact_365_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_366_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_367_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_left_mono
tff(fact_368_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_369_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_370_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_371_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% less_eqE
tff(fact_372_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_right_mono
tff(fact_373_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_374_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_375_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_376_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% le_iff_add
tff(fact_377_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_378_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_379_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_380_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_381_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_382_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_383_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_384_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_385_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_386_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_387_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_388_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K2: 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,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_389_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_390_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_391_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_392_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_393_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_394_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_395_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_396_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_397_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_398_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_strict_mono
tff(fact_399_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_400_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_401_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_strict_left_mono
tff(fact_402_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_403_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_404_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_405_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_strict_right_mono
tff(fact_406_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_407_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_408_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_409_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_410_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_411_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_412_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_413_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_414_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_415_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_416_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_417_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_418_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_419_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_420_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K2: 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,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_421_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K2: 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,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_422_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K2: 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,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_423_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K2: 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,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_424_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K2: 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,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_425_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_426_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_427_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_428_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_429_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_430_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_431_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_432_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_433_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
          & ? [M: nat] :
              ( ( some_nat(M) = vEBT_vebt_mint(Summarya) )
              & 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))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_434__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
    ( ( size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
    & aa_nat_o(vEBT_invar_vebt(summary),m) ) ).

% \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
tff(fact_435_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_436_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_437__C5_Ohyps_C_I11_J,axiom,
    ( ( 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))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I4 )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [X: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),na) = I4 )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),ma) ) ) ) ) ) ).

% "5.hyps"(11)
tff(fact_438_self__le__ge2__pow,axiom,
    ! [K2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)) ) ).

% self_le_ge2_pow
tff(fact_439_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_440_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_441_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_442_less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% less_exp
tff(fact_443_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_444_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_445_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => aa_real_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_446_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X4: complex] :
          ( member_complex(X4,set_complex2(Xs))
         => aa_complex_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_447_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,Nb: nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X4,set_Pr5648618587558075414at_nat(Xs))
         => aa_Pro6150637604445964947_nat_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s5460976970255530739at_nat(Xs))
       => aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,Nb)) ) ) ).

% inthall
tff(fact_448_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_449_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X4: $o] :
          ( member_o((X4),set_o2(Xs))
         => aa_o_o(P,(X4)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_450_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_nat_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_451_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => aa_int_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_452_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_453_high__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_454_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_455_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_456_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).

% add_diff_cancel_right'
tff(fact_457_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_458_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_459_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_right
tff(fact_460_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(A,B) ).

% add_diff_cancel_right
tff(fact_461_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_462_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_463_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_464_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),A) = B ).

% add_diff_cancel_left'
tff(fact_465_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_466_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_467_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).

% add_diff_cancel_left
tff(fact_468_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = minus_minus_rat(A,B) ).

% add_diff_cancel_left
tff(fact_469_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_470_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_471_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).

% diff_add_cancel
tff(fact_472_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),B) = A ).

% diff_add_cancel
tff(fact_473_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_474_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).

% add_diff_cancel
tff(fact_475_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),B) = A ).

% add_diff_cancel
tff(fact_476_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_477_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K2) ).

% Suc_diff_diff
tff(fact_478_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_479_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_480_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: 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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ).

% diff_diff_left
tff(fact_481_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_482_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),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)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_483_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_484_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_485_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_486_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))),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,K2),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_487_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_488_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Nb: 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,Nb) ) ).

% enat_less_induct
tff(fact_489_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_490_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_491_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_492_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) ).

% diff_right_commute
tff(fact_493_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : minus_minus_rat(minus_minus_rat(A,C),B) = minus_minus_rat(minus_minus_rat(A,B),C) ).

% diff_right_commute
tff(fact_494_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% diff_right_commute
tff(fact_495_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% diff_right_commute
tff(fact_496_diff__commute,axiom,
    ! [I: nat,J: nat,K2: 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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K2)),J) ).

% diff_commute
tff(fact_497_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_498_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_499_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_500_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Y: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_501_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_502_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_503_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_504_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_505_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_right_mono
tff(fact_506_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_right_mono
tff(fact_507_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_508_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_left_mono
tff(fact_509_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_left_mono
tff(fact_510_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_511_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_512_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_mono
tff(fact_513_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_514_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_strict_right_mono
tff(fact_515_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C)) ) ).

% diff_strict_right_mono
tff(fact_516_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_517_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_strict_left_mono
tff(fact_518_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B)) ) ).

% diff_strict_left_mono
tff(fact_519_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_520_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_521_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_522_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_523_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_524_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D)) ) ) ).

% diff_strict_mono
tff(fact_525_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_526_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_527_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),minus_minus_rat(C,D)) ).

% add_diff_add
tff(fact_528_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_529_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).

% diff_diff_eq
tff(fact_530_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_531_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_532_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_533_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).

% add_implies_diff
tff(fact_534_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = minus_minus_rat(A,B) ) ) ).

% add_implies_diff
tff(fact_535_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_536_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_537_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_538_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_539_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_540_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_add_eq
tff(fact_541_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).

% diff_add_eq
tff(fact_542_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_543_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).

% diff_diff_eq2
tff(fact_544_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C),B) ).

% diff_diff_eq2
tff(fact_545_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_546_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).

% add_diff_eq
tff(fact_547_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),C) ).

% add_diff_eq
tff(fact_548_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_549_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_550_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = minus_minus_rat(C,B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_551_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_552_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).

% diff_eq_eq
tff(fact_553_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat(A,B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_554_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_555_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_556_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( minus_minus_rat(A4,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_557_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_558_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K2: nat,I: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [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,K2),I)) ) ) ).

% zero_induct_lemma
tff(fact_559_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Nb)),K2) ) ).

% less_imp_diff_less
tff(fact_560_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_561_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_562_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_563_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_564_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_565_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_566_le__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_567_eq__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_568_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_569_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_570_diff__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_cancel2
tff(fact_571_Nat_Odiff__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_572_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_573_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_574_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_575_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_576_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_577_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_578_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_579_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_580_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_581_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_582_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_583_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% le_diff_eq
tff(fact_584_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_585_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_586_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_587_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_588_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_589_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_590_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_591_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_592_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),minus_minus_rat(C,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% less_diff_eq
tff(fact_593_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_594_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_595_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_596_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_597_less__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_598_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_599_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: 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),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ).

% less_diff_conv
tff(fact_600_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_601_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: 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) = K2 )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_602_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_603_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_604_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),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),K2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_605_le__diff__conv,axiom,
    ! [J: nat,K2: 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),K2)),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),K2)) ) ).

% le_diff_conv
tff(fact_606_power2__commute,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_607_power2__commute,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_608_power2__commute,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(minus_minus_rat(Y,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_609_power2__commute,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_610_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),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),K2)) ) ) ).

% less_diff_conv2
tff(fact_611_diff__le__diff__pow,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))) ) ).

% diff_le_diff_pow
tff(fact_612_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X4)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_613_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X4)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_614_power__divide,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_divide
tff(fact_615_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_616_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A,B)),Nb) = divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_617_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_618_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_619_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_620_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia != Maa )
       => ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_621_succ__list__to__short,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_622_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_623_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_624_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_625_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_626_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_627_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_628_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),minus_minus_rat(A,B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_629_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_630_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_631_low__inv,axiom,
    ! [Xb: nat,Nb: 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))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Xb ) ) ).

% low_inv
tff(fact_632_high__inv,axiom,
    ! [Xb: nat,Nb: 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))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Y ) ) ).

% high_inv
tff(fact_633_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Nb),set_real2(Xs)) ) ).

% nth_mem
tff(fact_634_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(aa_nat_complex(nth_complex(Xs),Nb),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_635_nth__mem,axiom,
    ! [Nb: nat,Xs: list_P6011104703257516679at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s5460976970255530739at_nat(Xs))
     => member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,Nb),set_Pr5648618587558075414at_nat(Xs)) ) ).

% nth_mem
tff(fact_636_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_637_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Nb),set_o2(Xs)) ) ).

% nth_mem
tff(fact_638_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_639_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(aa_nat_int(nth_int(Xs),Nb),set_int2(Xs)) ) ).

% nth_mem
tff(fact_640__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),na) ).

% \<open>1 \<le> n\<close>
tff(fact_641_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_642_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_643_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_644_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_645_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_646_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_647_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_648_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(Mb)),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_649_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_650_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_651_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_652_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_653_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_654_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_655_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_656_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_657_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.right_neutral
tff(fact_658_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_659_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_660_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_661_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_662_div__by__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).

% div_by_1
tff(fact_663_div__by__1,axiom,
    ! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).

% div_by_1
tff(fact_664_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_665_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_666_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_667_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% bits_div_by_1
tff(fact_668_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_669_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_670_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_671_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_672_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_673_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_674_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_675_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_676_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_677_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_678_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_679_not__Some__eq,axiom,
    ! [Xb: option_nat] :
      ( ! [Y5: nat] : Xb != some_nat(Y5)
    <=> ( Xb = none_nat ) ) ).

% not_Some_eq
tff(fact_680_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y5: num] : Xb != some_num(Y5)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_681_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_682_not__None__eq,axiom,
    ! [Xb: option_nat] :
      ( ( Xb != none_nat )
    <=> ? [Y5: nat] : Xb = some_nat(Y5) ) ).

% not_None_eq
tff(fact_683_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y5: num] : Xb = some_num(Y5) ) ).

% not_None_eq
tff(fact_684_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_685_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_686_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_687_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_688_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_689_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_690_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_691_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_692_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_693_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),numeral_numeral_rat(V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_694_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_695_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_696_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_697_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_698_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_699_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_700_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_701_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).

% left_diff_distrib_numeral
tff(fact_702_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V))) ).

% left_diff_distrib_numeral
tff(fact_703_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_704_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_705_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numeral_numeral_real(Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_706_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numeral_numeral_rat(Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_707_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_708_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_709_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_710_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = numeral_numeral_real(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_711_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = numeral_numeral_rat(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_712_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_713_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_714_power__inject__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_715_power__inject__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Mb) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_716_power__inject__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_717_power__inject__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),Mb) = aa_nat_int(power_power_int(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_718_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_719_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_720_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_721_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_722_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_723_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_724_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_725_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_726_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_727_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_728_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_729_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_730_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_731_power__strict__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_732_power__strict__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xb)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_733_power__strict__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_734_power__strict__increasing__iff,axiom,
    ! [B: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Xb)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_735_power__add__numeral2,axiom,
    ! [A: complex,Mb: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_736_power__add__numeral2,axiom,
    ! [A: real,Mb: num,Nb: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_737_power__add__numeral2,axiom,
    ! [A: rat,Mb: num,Nb: num,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_738_power__add__numeral2,axiom,
    ! [A: nat,Mb: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_739_power__add__numeral2,axiom,
    ! [A: int,Mb: num,Nb: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_740_power__add__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_741_power__add__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_742_power__add__numeral,axiom,
    ! [A: rat,Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_743_power__add__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_744_power__add__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_745_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_746_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_747_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_748_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_749_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_750_power__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_751_power__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xb)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_752_power__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_753_power__increasing__iff,axiom,
    ! [B: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Xb)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_754_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_755_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Va: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Mia)
         => ( ( Maa != Mia )
           => 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(Summarya))),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(TreeLista,the_nat(vEBT_vebt_mint(Summarya))))))),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(TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_756_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_757_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),one_one_real) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_758_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Nb)),one_one_rat) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_759_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_760_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_761_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_762_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Nb)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_763_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(Nb)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_764_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_765_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_766_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_767_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_768_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_769_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_770_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),numeral_numeral_real(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_771_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),numeral_numeral_rat(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_772_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_773_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_774_less__1__mult,axiom,
    ! [Mb: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_775_less__1__mult,axiom,
    ! [Mb: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Mb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_776_less__1__mult,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_777_less__1__mult,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_778_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_779_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_780_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_781_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_782_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_783_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_784_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_785_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_786_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_787_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_788_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.left_commute
tff(fact_789_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.left_commute
tff(fact_790_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_791_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_792_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_793_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_794_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_795_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_796_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_797_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).

% mult.commute
tff(fact_798_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A) ).

% mult.commute
tff(fact_799_mult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% mult.commute
tff(fact_800_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% mult.commute
tff(fact_801_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% mult.assoc
tff(fact_802_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% mult.assoc
tff(fact_803_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_804_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_805_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_806_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_807_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_808_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_809_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_810_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_811_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_812_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_813_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_814_left__right__inverse__power,axiom,
    ! [Xb: complex,Y: complex,Nb: 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),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_815_left__right__inverse__power,axiom,
    ! [Xb: real,Y: real,Nb: 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),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_816_left__right__inverse__power,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_817_left__right__inverse__power,axiom,
    ! [Xb: nat,Y: nat,Nb: 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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_818_left__right__inverse__power,axiom,
    ! [Xb: int,Y: int,Nb: 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),Nb)),aa_nat_int(power_power_int(Y),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_819_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_820_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_821_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_822_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_823_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_824_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_825_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_826_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_827_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_828_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% comm_semiring_class.distrib
tff(fact_829_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_830_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_831_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_832_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_833_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% distrib_left
tff(fact_834_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_835_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_836_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% distrib_right
tff(fact_837_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% distrib_right
tff(fact_838_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_839_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_840_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E)),C) ).

% combine_common_factor
tff(fact_841_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_842_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_843_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).

% combine_common_factor
tff(fact_844_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% left_diff_distrib
tff(fact_845_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% left_diff_distrib
tff(fact_846_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_847_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_848_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib
tff(fact_849_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_850_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_851_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)) ).

% left_diff_distrib'
tff(fact_852_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_853_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_854_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_855_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib'
tff(fact_856_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_857_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_858_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_859_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_860_square__diff__one__factored,axiom,
    ! [Xb: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)),minus_minus_rat(Xb,one_one_rat)) ).

% square_diff_one_factored
tff(fact_861_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_862_power__less__power__Suc,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_863_power__less__power__Suc,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_864_power__less__power__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_865_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_866_power__gt1__lemma,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_867_power__gt1__lemma,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_868_power__gt1__lemma,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_869_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_870_power__commutes,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_commutes
tff(fact_871_power__commutes,axiom,
    ! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_commutes
tff(fact_872_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_873_power__commutes,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_commutes
tff(fact_874_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_commutes
tff(fact_875_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_mult_distrib
tff(fact_876_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_877_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_878_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_879_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ).

% power_mult_distrib
tff(fact_880_power__commuting__commutes,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_881_power__commuting__commutes,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_882_power__commuting__commutes,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xb) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Nb)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_883_power__commuting__commutes,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xb) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_884_power__commuting__commutes,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Nb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_885_Suc__mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_886_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc1260151622200656695at_nat] :
      ( ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : Xb != produc7814092408141220007at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_887_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc7004362771917097900on_nat] :
      ( ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : Xb != produc6699827698519632094on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_nat,V2: nat] : Xb != produc6699827698519632094on_nat(Uw,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat] : Xb != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A3),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_888_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc2703417854250991842on_num] :
      ( ! [Uu: fun_num_fun_num_num,Uv: option_num] : Xb != produc6360833703471310868on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_num,V2: num] : Xb != produc6360833703471310868on_num(Uw,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A3: num,B3: num] : Xb != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A3),some_num(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_889_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc3739950874144486240at_nat] :
      ( ! [Uu: fun_Pr2028404973800837934_nat_o,Uv: option4927543243414619207at_nat] : Xb != produc3653971535690468304at_nat(Uu,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2028404973800837934_nat_o,X4: product_prod_nat_nat,Y3: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X4),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_890_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc1615844007061411358on_nat] :
      ( ! [Uu: fun_nat_fun_nat_o,Uv: option_nat] : Xb != produc7967433646104525454on_nat(Uu,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_o,V2: nat] : Xb != produc7967433646104525454on_nat(Uw,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X4: nat,Y3: nat] : Xb != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X4),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_891_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc50614890452615238on_num] :
      ( ! [Uu: fun_num_fun_num_o,Uv: option_num] : Xb != produc1140783415258738358on_num(Uu,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_o,V2: num] : Xb != produc1140783415258738358on_num(Uw,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X4: num,Y3: num] : Xb != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X4),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_892_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) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => 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_893_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) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => 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_894_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) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B3) )
               => 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_895_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) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( Xb = some_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => 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_896_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) )
       => ( ! [A3: nat,B3: nat] :
              ( ( Xb = some_nat(A3) )
             => ( ( Y = some_nat(B3) )
               => 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_897_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) )
       => ( ! [A3: nat,B3: num] :
              ( ( Xb = some_nat(A3) )
             => ( ( Y = some_num(B3) )
               => 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_898_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) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B3) )
               => 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_899_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) )
       => ( ! [A3: num,B3: nat] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_nat(B3) )
               => 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_900_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) )
       => ( ! [A3: num,B3: num] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_num(B3) )
               => 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_901_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)
        & ! [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_all
tff(fact_902_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)
        & ! [X3: nat] : aa_option_nat_o(P,some_nat(X3)) ) ) ).

% split_option_all
tff(fact_903_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)
        & ! [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_all
tff(fact_904_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)
        | ? [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_ex
tff(fact_905_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)
        | ? [X3: nat] : aa_option_nat_o(P,some_nat(X3)) ) ) ).

% split_option_ex
tff(fact_906_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)
        | ? [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_ex
tff(fact_907_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).

% option.exhaust
tff(fact_908_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X22: nat] : Y != some_nat(X22) ) ).

% option.exhaust
tff(fact_909_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] : Y != some_num(X22) ) ).

% option.exhaust
tff(fact_910_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_911_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_912_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_913_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_914_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_915_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_916_power__mult,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),Nb) ).

% power_mult
tff(fact_917_power__mult,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),Nb) ).

% power_mult
tff(fact_918_power__mult,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Mb)),Nb) ).

% power_mult
tff(fact_919_power__mult,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Mb)),Nb) ).

% power_mult
tff(fact_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_le__cube,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).

% le_cube
tff(fact_929_le__square,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).

% le_square
tff(fact_930_mult__le__mono,axiom,
    ! [I: nat,J: nat,K2: 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,K2),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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_931_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ).

% mult_le_mono1
tff(fact_932_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K2: 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,K2),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ).

% mult_le_mono2
tff(fact_933_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% add_mult_distrib
tff(fact_934_add__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% add_mult_distrib2
tff(fact_935_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K2) ).

% left_add_mult_distrib
tff(fact_936_diff__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% diff_mult_distrib2
tff(fact_937_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% diff_mult_distrib
tff(fact_938_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q3) ).

% div_mult2_eq
tff(fact_939_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_940_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_941_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_942_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_943_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_944_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_945_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_946_square__diff__square__factored,axiom,
    ! [Xb: rat,Y: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)),minus_minus_rat(Xb,Y)) ).

% square_diff_square_factored
tff(fact_947_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_948_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_949_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_950_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_951_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_952_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_953_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_954_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_955_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_956_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X2)) = X2 ).

% option.sel
tff(fact_957_option_Osel,axiom,
    ! [X2: nat] : the_nat(some_nat(X2)) = X2 ).

% option.sel
tff(fact_958_option_Osel,axiom,
    ! [X2: num] : the_num(some_num(X2)) = X2 ).

% option.sel
tff(fact_959_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_960_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu2,none_P5556105721700978146at_nat,Uv2) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_961_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : vEBT_V819420779217536731ft_num(Uu2,none_num,Uv2) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_962_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu2),none_nat),Uv2) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_963_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_964_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_965_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_966_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_967_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_968_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_969_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_970_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B,A)),E)),D)) ) ).

% less_add_iff2
tff(fact_971_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_972_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_973_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_974_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_975_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_976_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(one)),A) = A ).

% mult_numeral_1
tff(fact_977_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(one)),A) = A ).

% mult_numeral_1
tff(fact_978_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_979_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_980_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_981_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(one)) = A ).

% mult_numeral_1_right
tff(fact_982_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(one)) = A ).

% mult_numeral_1_right
tff(fact_983_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_984_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_985_mult__diff__mult,axiom,
    ! [Xb: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_986_mult__diff__mult,axiom,
    ! [Xb: rat,Y: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),minus_minus_rat(Y,B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xb,A)),B)) ).

% mult_diff_mult
tff(fact_987_mult__diff__mult,axiom,
    ! [Xb: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_988_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) ).

% power_Suc2
tff(fact_989_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).

% power_Suc2
tff(fact_990_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_991_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_992_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_993_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_Suc
tff(fact_994_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_Suc
tff(fact_995_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_996_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_997_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_998_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_999_power__add,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_1000_power__add,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_1001_power__add,axiom,
    ! [A: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_1002_power__add,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_1003_power__add,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_1004_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),numeral_numeral_real(Nb)) ).

% one_le_numeral
tff(fact_1005_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(Nb)) ).

% one_le_numeral
tff(fact_1006_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_1007_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_1008_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_1009_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_1010_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_1011_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_1012_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_1013_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_1014_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_1015_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_1016_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(Xb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Xb)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_1017_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_1018_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_1019_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_1020_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_1021_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_1022_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_1023_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_1024_one__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% one_le_power
tff(fact_1025_one__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% one_le_power
tff(fact_1026_one__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% one_le_power
tff(fact_1027_one__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% one_le_power
tff(fact_1028_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_1029_power__one__over,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),Nb) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_one_over
tff(fact_1030_power__one__over,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_one_over
tff(fact_1031_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),Nb) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_1032_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_1033_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_1034_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_1035_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_1036_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1037_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1038_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_1039_power__odd__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1040_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1041_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1042_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1043_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1044_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 ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A3) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B3) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1045_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 ) ) )
         => ~ ! [A3: num] :
                ( ( Xaa = some_num(A3) )
               => ! [B3: num] :
                    ( ( Xba = some_num(B3) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(Xb,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1046_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 ) ) )
         => ~ ! [A3: nat] :
                ( ( Xaa = some_nat(A3) )
               => ! [B3: nat] :
                    ( ( Xba = some_nat(B3) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1047_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw2,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1048_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw2,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1049_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw2),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1050_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_1051_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_1052_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_1053_power__gt1,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1054_power__gt1,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1055_power__gt1,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1056_power__gt1,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_1057_power__less__imp__less__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1058_power__less__imp__less__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1059_power__less__imp__less__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1060_power__less__imp__less__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_1061_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1062_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1063_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1064_power__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N3)) ) ) ).

% power_strict_increasing
tff(fact_1065_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_1066_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1067_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_1068_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_1069_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_1070_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_1071_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)) ) ) ).

% power_increasing
tff(fact_1072_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N3)) ) ) ).

% power_increasing
tff(fact_1073_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)) ) ) ).

% power_increasing
tff(fact_1074_power__increasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(A),N3)) ) ) ).

% power_increasing
tff(fact_1075_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B4)
    <=> ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => member_real(X3,B4) ) ) ).

% subset_code(1)
tff(fact_1076_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B4: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B4)
    <=> ! [X3: complex] :
          ( member_complex(X3,set_complex2(Xs))
         => member_complex(X3,B4) ) ) ).

% subset_code(1)
tff(fact_1077_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs),B4)
    <=> ! [X3: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X3,set_Pr5648618587558075414at_nat(Xs))
         => member8440522571783428010at_nat(X3,B4) ) ) ).

% subset_code(1)
tff(fact_1078_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B4)
    <=> ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X3,B4) ) ) ).

% subset_code(1)
tff(fact_1079_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B4)
    <=> ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => member_int(X3,B4) ) ) ).

% subset_code(1)
tff(fact_1080_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B4)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X3),B4) ) ) ).

% subset_code(1)
tff(fact_1081_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1082_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1083_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1084_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1085_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1086_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1087_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1088_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1089_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_1090_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_1091_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2
tff(fact_1092_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_1093_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_1094_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_1095_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),numeral_numeral_real(aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_1096_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),numeral_numeral_rat(aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2_right
tff(fact_1097_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_1098_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_1099_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_1100_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_1101_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_1102_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_1103_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_1104_power4__eq__xxxx,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_1105_power4__eq__xxxx,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_1106_power4__eq__xxxx,axiom,
    ! [Xb: rat] : aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_1107_power4__eq__xxxx,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_1108_power4__eq__xxxx,axiom,
    ! [Xb: int] : aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_1109_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_1110_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_1111_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% power2_eq_square
tff(fact_1112_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_1113_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_1114_power__even__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_1115_power__even__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_1116_power__even__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_1117_power__even__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_1118_div__nat__eqI,axiom,
    ! [Nb: nat,Q3: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_1119_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1120_power__le__imp__le__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1121_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1122_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_1123_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_1124_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_1125_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_1126_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_1127_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_1128_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_1129_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_1130_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_1131_power2__sum,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_1132_power2__sum,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_1133_power2__sum,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(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_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_1134_power2__sum,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,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_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_1135_power2__sum,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(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_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_1136_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_1137_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_1138_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_1139_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_1140_power2__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_1141_power2__diff,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_1142_power2__diff,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(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_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_1143_power2__diff,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(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_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_1144_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K2)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_1145_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_1146_add__le__imp__le__diff,axiom,
    ! [I: real,K2: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_1147_add__le__imp__le__diff,axiom,
    ! [I: rat,K2: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),minus_minus_rat(Nb,K2)) ) ).

% add_le_imp_le_diff
tff(fact_1148_add__le__imp__le__diff,axiom,
    ! [I: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_1149_add__le__imp__le__diff,axiom,
    ! [I: int,K2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_1150_add__le__add__imp__diff__le,axiom,
    ! [I: real,K2: real,Nb: 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),K2)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K2)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K2))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1151_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K2: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K2))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K2)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K2))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Nb,K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1152_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K2: nat,Nb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1153_add__le__add__imp__diff__le,axiom,
    ! [I: int,K2: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K2)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1154_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1155_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),minus_minus_rat(A,B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1156_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1157_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1158_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1159_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1160_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1161_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1162_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I3),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K2 )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I3),nth_VEBT_VEBT(Xs3,I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_1163_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_o_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I3),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K2 )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => aa_o_o(aa_nat_fun_o_o(P,I3),aa_nat_o(nth_o(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_1164_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K2 )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(nth_nat(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_1165_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_int_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I3),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K2 )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => aa_int_o(aa_nat_fun_int_o(P,I3),aa_nat_int(nth_int(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_1166_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1167_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1168_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1169_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1170_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ! [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_1171_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X3: $o] :
          ( member_o((X3),set_o2(Xs))
         => aa_o_o(P,(X3)) )
    <=> ! [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_1172_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
    <=> ! [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_1173_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => aa_int_o(P,X3) )
    <=> ! [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_1174_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_1175_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I2)) )
     => ( member_complex(Xb,set_complex2(Xs))
       => aa_complex_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1176_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_1177_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_1178_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_1179_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_1180_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) )
     => ( member_int(Xb,set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1181_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_1182_in__set__conv__nth,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,set_complex2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1183_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_1184_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_1185_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_1186_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_1187_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( 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_1188_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X4) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_1189_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X4: $o] :
            ( member_o((X4),set_o2(Xs))
           => aa_o_o(P,(X4)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1190_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
           => aa_nat_o(P,X4) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1191_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X4: int] :
            ( member_int(X4,set_int2(Xs))
           => aa_int_o(P,X4) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1192_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_1193_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_1194_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,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_1195_sum__squares__bound,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)),aa_rat_rat(aa_rat_fun_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_1196_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_1197_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_1198_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_1199_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_1200_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_1201_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).

% times_divide_eq_left
tff(fact_1202_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A),C) ).

% times_divide_eq_left
tff(fact_1203_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_1204_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).

% divide_divide_eq_left
tff(fact_1205_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% divide_divide_eq_left
tff(fact_1206_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_1207_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).

% divide_divide_eq_right
tff(fact_1208_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),B) ).

% divide_divide_eq_right
tff(fact_1209_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_1210_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).

% times_divide_eq_right
tff(fact_1211_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),C) ).

% times_divide_eq_right
tff(fact_1212_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_13) ) ).

% not_min_Null_member
tff(fact_1213_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_1214_semiring__norm_I13_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% semiring_norm(13)
tff(fact_1215_semiring__norm_I11_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Mb),one) = Mb ).

% semiring_norm(11)
tff(fact_1216_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_1217_num__double,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_1218_power__mult__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_1219_power__mult__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_1220_power__mult__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_1221_power__mult__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_1222_four__x__squared,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_1223_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1224_div__mult2__numeral__eq,axiom,
    ! [A: nat,K2: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K2),L))) ).

% div_mult2_numeral_eq
tff(fact_1225_div__mult2__numeral__eq,axiom,
    ! [A: int,K2: num,L: num] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K2),L))) ).

% div_mult2_numeral_eq
tff(fact_1226_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_1227_linordered__field__no__ub,axiom,
    ! [X: real] :
    ? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),X_13) ).

% linordered_field_no_ub
tff(fact_1228_linordered__field__no__ub,axiom,
    ! [X: rat] :
    ? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_13) ).

% linordered_field_no_ub
tff(fact_1229_linordered__field__no__lb,axiom,
    ! [X: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X) ).

% linordered_field_no_lb
tff(fact_1230_linordered__field__no__lb,axiom,
    ! [X: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X) ).

% linordered_field_no_lb
tff(fact_1231_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_1232_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1233_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1234_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_1235_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).

% divide_divide_eq_left'
tff(fact_1236_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ).

% divide_divide_eq_left'
tff(fact_1237_divide__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),W)),aa_complex_complex(times_times_complex(Y),Z)) ).

% divide_divide_times_eq
tff(fact_1238_divide__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_1239_divide__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(Xb,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),W),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ).

% divide_divide_times_eq
tff(fact_1240_times__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_1241_times__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_1242_times__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),W)) ).

% times_divide_times_eq
tff(fact_1243_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_1244_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% add_divide_distrib
tff(fact_1245_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ).

% add_divide_distrib
tff(fact_1246_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_1247_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).

% diff_divide_distrib
tff(fact_1248_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : divide_divide_rat(minus_minus_rat(A,B),C) = minus_minus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).

% diff_divide_distrib
tff(fact_1249_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1250_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1251_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_1252_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_1253_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_1254_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_1255_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_1256_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_1257_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_1258_succ__less__length__list,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),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,Dega),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_1259_succ__greatereq__min,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: 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))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),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,Dega),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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      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,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_1260_divmod__step__eq,axiom,
    ! [L: num,Q3: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),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))),Q3)),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))),Q3)),R2)) ).

% divmod_step_eq
tff(fact_1261_divmod__step__eq,axiom,
    ! [L: num,Q3: int,R2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),R2)) ).

% divmod_step_eq
tff(fact_1262_divmod__step__eq,axiom,
    ! [L: num,Q3: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q3),R2)) ).

% divmod_step_eq
tff(fact_1263_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1264_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1265_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_1266_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_1267_low__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_1268_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_1269_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1270_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_1271_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mod_trivial
tff(fact_1272_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_1273_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self1
tff(fact_1274_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_1275_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_add_self2
tff(fact_1276_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mod_self2
tff(fact_1277_real__divide__square__eq,axiom,
    ! [R2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R2) ).

% real_divide_square_eq
tff(fact_1278_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = Mb ) ) ).

% mod_less
tff(fact_1279_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_1280_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self4
tff(fact_1281_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_1282_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self3
tff(fact_1283_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_1284_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self2
tff(fact_1285_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_1286_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(modulo_modulo_int(A),B) ).

% mod_mult_self1
tff(fact_1287_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K2: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_1288_Suc__mod__mult__self3,axiom,
    ! [K2: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_1289_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_1290_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_1291_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_1292_one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_1293_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_1294_bits__one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_1295_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_1296_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) != 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,K2)),Nb))),aa_num_nat(numeral_numeral_nat,K2)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1297_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_1298_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_ag,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_1299_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_ah(C) = aa_rat_fun_rat_rat(times_times_rat,C) ).

% mult_commute_abs
tff(fact_1300_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ai,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_1301_mult__commute__abs,axiom,
    ! [C: int] : aTP_Lamm_aj(C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_1302_lambda__one,axiom,
    aTP_Lamm_ak = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_1303_lambda__one,axiom,
    aTP_Lamm_al = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_1304_lambda__one,axiom,
    aTP_Lamm_am = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_1305_lambda__one,axiom,
    aTP_Lamm_an = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1306_lambda__one,axiom,
    aTP_Lamm_ao = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_1307_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_1308_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_right_eq
tff(fact_1309_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_1310_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_left_eq
tff(fact_1311_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_1312_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).

% mult_mod_right
tff(fact_1313_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_1314_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),B)),C) ).

% mod_mult_mult2
tff(fact_1315_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1316_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A2),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B2),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1317_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_1318_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_1319_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_1320_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_1321_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) ) ) ) ).

% mod_add_cong
tff(fact_1322_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A2),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B2),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ) ) ) ).

% mod_add_cong
tff(fact_1323_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_1324_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_1325_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_1326_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_1327_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_right_eq
tff(fact_1328_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),C)),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_left_eq
tff(fact_1329_mod__diff__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A2),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B2),C) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) ) ) ) ).

% mod_diff_cong
tff(fact_1330_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_1331_power__mod,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_1332_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A),B)),Nb)),B) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A),Nb)),B) ).

% power_mod
tff(fact_1333_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_1334_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_1335_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_1336_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1337_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1338_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1339_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1340_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m2),m2) ) ).

% numeral_code(2)
tff(fact_1341_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= numeral_numeral_real(Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2) ) ).

% numeral_code(2)
tff(fact_1342_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numeral_numeral_rat(aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= numeral_numeral_rat(Nb),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2) ) ).

% numeral_code(2)
tff(fact_1343_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_1344_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_1345_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_1346_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_1347_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1348_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_1349_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_1350_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).

% power_numeral_even
tff(fact_1351_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_1352_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_1353_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_1354_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_1355_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_1356_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_1357_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
     => ~ ! [D2: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ).

% mod_eqE
tff(fact_1358_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_1359_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),C)),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_add1_eq
tff(fact_1360_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
           => aa_nat_o(P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_1361_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_1362_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ~ ! [S: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).

% mod_eq_nat1E
tff(fact_1363_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S)) ) ) ).

% mod_eq_nat2E
tff(fact_1364_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ? [Q2: 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,Nb),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1365_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ).

% mod_if
tff(fact_1366_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_1367_set__conv__nth,axiom,
    ! [Xs: list_complex] : set_complex2(Xs) = collect_complex(aTP_Lamm_ap(Xs)) ).

% set_conv_nth
tff(fact_1368_set__conv__nth,axiom,
    ! [Xs: list_real] : set_real2(Xs) = collect_real(aTP_Lamm_aq(Xs)) ).

% set_conv_nth
tff(fact_1369_set__conv__nth,axiom,
    ! [Xs: list_list_nat] : set_list_nat2(Xs) = collect_list_nat(aTP_Lamm_ar(Xs)) ).

% set_conv_nth
tff(fact_1370_set__conv__nth,axiom,
    ! [Xs: list_set_nat] : set_set_nat2(Xs) = collect_set_nat(aTP_Lamm_as(Xs)) ).

% set_conv_nth
tff(fact_1371_set__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs) = collect_VEBT_VEBT(aTP_Lamm_at(Xs)) ).

% set_conv_nth
tff(fact_1372_set__conv__nth,axiom,
    ! [Xs: list_o] : set_o2(Xs) = collect_o(aTP_Lamm_au(Xs)) ).

% set_conv_nth
tff(fact_1373_set__conv__nth,axiom,
    ! [Xs: list_nat] : set_nat2(Xs) = collect_nat(aTP_Lamm_av(Xs)) ).

% set_conv_nth
tff(fact_1374_set__conv__nth,axiom,
    ! [Xs: list_int] : set_int2(Xs) = collect_int(aTP_Lamm_aw(Xs)) ).

% set_conv_nth
tff(fact_1375_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1376_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1377_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1378_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1379_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_1380_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) ).

% mod_div_decomp
tff(fact_1381_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_1382_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)),aa_int_int(modulo_modulo_int(A),B)) = A ).

% div_mult_mod_eq
tff(fact_1383_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1384_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_1385_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_1386_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))) = A ).

% mod_mult_div_eq
tff(fact_1387_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_1388_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))),aa_int_int(modulo_modulo_int(A),B)) = A ).

% mult_div_mod_eq
tff(fact_1389_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_1390_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(modulo_modulo_int(B),C))),C)) ).

% div_mult1_eq
tff(fact_1391_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_1392_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B)) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_div_mult_eq_mod
tff(fact_1393_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_1394_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_1395_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_1396_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B)) ).

% minus_mod_eq_mult_div
tff(fact_1397_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_1398_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(A),B))) = aa_int_int(modulo_modulo_int(A),B) ).

% minus_mult_div_eq_mod
tff(fact_1399_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_1400_modulo__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_1401_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1402_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Mb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1403_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1404_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1405_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
        some_nat(Mia),
        $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(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  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_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_1406_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_1407_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1408_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A4,B4) = bot_bot_set_int )
    <=> ord_less_eq_set_int(A4,B4) ) ).

% Diff_eq_empty_iff
tff(fact_1409_Diff__eq__empty__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( minus_minus_set_real(A4,B4) = bot_bot_set_real )
    <=> ord_less_eq_set_real(A4,B4) ) ).

% Diff_eq_empty_iff
tff(fact_1410_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat(A4,B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1411_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_1412_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S2)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_1413_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_1414_div__mod__decomp,axiom,
    ! [A4: nat,Nb: nat] : A4 = 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,A4),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),Nb)) ).

% div_mod_decomp
tff(fact_1415_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1416_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1417_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1418_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1419_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1420_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1421_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_1422_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_1423_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_1424_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_1425_Diff__idemp,axiom,
    ! [A4: set_nat,B4: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A4,B4),B4) = minus_minus_set_nat(A4,B4) ).

% Diff_idemp
tff(fact_1426_Diff__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( member_int(C,minus_minus_set_int(A4,B4))
    <=> ( member_int(C,A4)
        & ~ member_int(C,B4) ) ) ).

% Diff_iff
tff(fact_1427_Diff__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( member_real(C,minus_minus_set_real(A4,B4))
    <=> ( member_real(C,A4)
        & ~ member_real(C,B4) ) ) ).

% Diff_iff
tff(fact_1428_Diff__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A4,B4))
    <=> ( member_complex(C,A4)
        & ~ member_complex(C,B4) ) ) ).

% Diff_iff
tff(fact_1429_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,minus_1356011639430497352at_nat(A4,B4))
    <=> ( member8440522571783428010at_nat(C,A4)
        & ~ member8440522571783428010at_nat(C,B4) ) ) ).

% Diff_iff
tff(fact_1430_Diff__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A4,B4))
    <=> ( aa_set_nat_o(member_nat(C),A4)
        & ~ aa_set_nat_o(member_nat(C),B4) ) ) ).

% Diff_iff
tff(fact_1431_DiffI,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( member_int(C,A4)
     => ( ~ member_int(C,B4)
       => member_int(C,minus_minus_set_int(A4,B4)) ) ) ).

% DiffI
tff(fact_1432_DiffI,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( member_real(C,A4)
     => ( ~ member_real(C,B4)
       => member_real(C,minus_minus_set_real(A4,B4)) ) ) ).

% DiffI
tff(fact_1433_DiffI,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( member_complex(C,A4)
     => ( ~ member_complex(C,B4)
       => member_complex(C,minus_811609699411566653omplex(A4,B4)) ) ) ).

% DiffI
tff(fact_1434_DiffI,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,A4)
     => ( ~ member8440522571783428010at_nat(C,B4)
       => member8440522571783428010at_nat(C,minus_1356011639430497352at_nat(A4,B4)) ) ) ).

% DiffI
tff(fact_1435_DiffI,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),A4)
     => ( ~ aa_set_nat_o(member_nat(C),B4)
       => aa_set_nat_o(member_nat(C),minus_minus_set_nat(A4,B4)) ) ) ).

% DiffI
tff(fact_1436_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xb)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_1437_Diff__empty,axiom,
    ! [A4: set_int] : minus_minus_set_int(A4,bot_bot_set_int) = A4 ).

% Diff_empty
tff(fact_1438_Diff__empty,axiom,
    ! [A4: set_real] : minus_minus_set_real(A4,bot_bot_set_real) = A4 ).

% Diff_empty
tff(fact_1439_Diff__empty,axiom,
    ! [A4: set_nat] : minus_minus_set_nat(A4,bot_bot_set_nat) = A4 ).

% Diff_empty
tff(fact_1440_empty__Diff,axiom,
    ! [A4: set_int] : minus_minus_set_int(bot_bot_set_int,A4) = bot_bot_set_int ).

% empty_Diff
tff(fact_1441_empty__Diff,axiom,
    ! [A4: set_real] : minus_minus_set_real(bot_bot_set_real,A4) = bot_bot_set_real ).

% empty_Diff
tff(fact_1442_empty__Diff,axiom,
    ! [A4: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A4) = bot_bot_set_nat ).

% empty_Diff
tff(fact_1443_Diff__cancel,axiom,
    ! [A4: set_int] : minus_minus_set_int(A4,A4) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1444_Diff__cancel,axiom,
    ! [A4: set_real] : minus_minus_set_real(A4,A4) = bot_bot_set_real ).

% Diff_cancel
tff(fact_1445_Diff__cancel,axiom,
    ! [A4: set_nat] : minus_minus_set_nat(A4,A4) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1446_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_1447_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ord_less_set_int(A4,B4)
     => ? [B3: int] : member_int(B3,minus_minus_set_int(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1448_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ord_less_set_real(A4,B4)
     => ? [B3: real] : member_real(B3,minus_minus_set_real(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1449_psubset__imp__ex__mem,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ord_less_set_complex(A4,B4)
     => ? [B3: complex] : member_complex(B3,minus_811609699411566653omplex(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1450_psubset__imp__ex__mem,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A4,B4)
     => ? [B3: product_prod_nat_nat] : member8440522571783428010at_nat(B3,minus_1356011639430497352at_nat(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1451_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A4),B4)
     => ? [B3: nat] : aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1452_minus__set__def,axiom,
    ! [A4: set_int,B4: set_int] : minus_minus_set_int(A4,B4) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ).

% minus_set_def
tff(fact_1453_minus__set__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A4,B4) = collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(minus_2386458686304230003_nat_o(aTP_Lamm_aa(A4)),aTP_Lamm_aa(B4))) ).

% minus_set_def
tff(fact_1454_minus__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] : minus_811609699411566653omplex(A4,B4) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ).

% minus_set_def
tff(fact_1455_minus__set__def,axiom,
    ! [A4: set_real,B4: set_real] : minus_minus_set_real(A4,B4) = collect_real(minus_404528632214381484real_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ).

% minus_set_def
tff(fact_1456_minus__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ).

% minus_set_def
tff(fact_1457_minus__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ).

% minus_set_def
tff(fact_1458_minus__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : minus_minus_set_nat(A4,B4) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_af(A4),aTP_Lamm_af(B4))) ).

% minus_set_def
tff(fact_1459_set__diff__eq,axiom,
    ! [A4: set_int,B4: set_int] : minus_minus_set_int(A4,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_ax(A4),B4)) ).

% set_diff_eq
tff(fact_1460_set__diff__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A4,B4) = collec3392354462482085612at_nat(aa_set1857767853180004543_nat_o(aTP_Lamm_ay(A4),B4)) ).

% set_diff_eq
tff(fact_1461_set__diff__eq,axiom,
    ! [A4: set_complex,B4: set_complex] : minus_811609699411566653omplex(A4,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_az(A4),B4)) ).

% set_diff_eq
tff(fact_1462_set__diff__eq,axiom,
    ! [A4: set_real,B4: set_real] : minus_minus_set_real(A4,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_ba(A4),B4)) ).

% set_diff_eq
tff(fact_1463_set__diff__eq,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_bb(A4),B4)) ).

% set_diff_eq
tff(fact_1464_set__diff__eq,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_bc(A4),B4)) ).

% set_diff_eq
tff(fact_1465_set__diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] : minus_minus_set_nat(A4,B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bd(A4),B4)) ).

% set_diff_eq
tff(fact_1466_DiffD2,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( member_int(C,minus_minus_set_int(A4,B4))
     => ~ member_int(C,B4) ) ).

% DiffD2
tff(fact_1467_DiffD2,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( member_real(C,minus_minus_set_real(A4,B4))
     => ~ member_real(C,B4) ) ).

% DiffD2
tff(fact_1468_DiffD2,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A4,B4))
     => ~ member_complex(C,B4) ) ).

% DiffD2
tff(fact_1469_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,minus_1356011639430497352at_nat(A4,B4))
     => ~ member8440522571783428010at_nat(C,B4) ) ).

% DiffD2
tff(fact_1470_DiffD2,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A4,B4))
     => ~ aa_set_nat_o(member_nat(C),B4) ) ).

% DiffD2
tff(fact_1471_DiffD1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( member_int(C,minus_minus_set_int(A4,B4))
     => member_int(C,A4) ) ).

% DiffD1
tff(fact_1472_DiffD1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( member_real(C,minus_minus_set_real(A4,B4))
     => member_real(C,A4) ) ).

% DiffD1
tff(fact_1473_DiffD1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A4,B4))
     => member_complex(C,A4) ) ).

% DiffD1
tff(fact_1474_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,minus_1356011639430497352at_nat(A4,B4))
     => member8440522571783428010at_nat(C,A4) ) ).

% DiffD1
tff(fact_1475_DiffD1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A4,B4))
     => aa_set_nat_o(member_nat(C),A4) ) ).

% DiffD1
tff(fact_1476_DiffE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( member_int(C,minus_minus_set_int(A4,B4))
     => ~ ( member_int(C,A4)
         => member_int(C,B4) ) ) ).

% DiffE
tff(fact_1477_DiffE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( member_real(C,minus_minus_set_real(A4,B4))
     => ~ ( member_real(C,A4)
         => member_real(C,B4) ) ) ).

% DiffE
tff(fact_1478_DiffE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A4,B4))
     => ~ ( member_complex(C,A4)
         => member_complex(C,B4) ) ) ).

% DiffE
tff(fact_1479_DiffE,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,minus_1356011639430497352at_nat(A4,B4))
     => ~ ( member8440522571783428010at_nat(C,A4)
         => member8440522571783428010at_nat(C,B4) ) ) ).

% DiffE
tff(fact_1480_DiffE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A4,B4))
     => ~ ( aa_set_nat_o(member_nat(C),A4)
         => aa_set_nat_o(member_nat(C),B4) ) ) ).

% DiffE
tff(fact_1481_div__mod__decomp__int,axiom,
    ! [A4: int,Nb: int] : A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A4),Nb)),Nb)),aa_int_int(modulo_modulo_int(A4),Nb)) ).

% div_mod_decomp_int
tff(fact_1482_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_1483_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_1484_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_1485_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_1486_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% verit_comp_simplify1(2)
tff(fact_1487_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1488_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1489_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1490_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1491_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1492_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1493_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1494_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1495_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1496_double__diff,axiom,
    ! [A4: set_nat,B4: set_nat,C4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C4)
       => ( minus_minus_set_nat(B4,minus_minus_set_nat(C4,A4)) = A4 ) ) ) ).

% double_diff
tff(fact_1497_Diff__subset,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A4,B4)),A4) ).

% Diff_subset
tff(fact_1498_Diff__mono,axiom,
    ! [A4: set_nat,C4: set_nat,D3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),C4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D3),B4)
       => aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A4,B4)),minus_minus_set_nat(C4,D3)) ) ) ).

% Diff_mono
tff(fact_1499_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1500_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1501_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1502_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1503_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1504_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_1505_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)),Xb) ).

% vebt_member.simps(2)
tff(fact_1506_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_1507_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1508_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_1509_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((Uu),(B3)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y != none_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            some_nat(Mi),
                            $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,Va3))),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,Va3))),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(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1510_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_1511_product__nth,axiom,
    ! [Nb: nat,Xs: list_Code_integer,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys),Nb) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_1512_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_1513_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_1514_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_1515_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_1516_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_1517_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Nb) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_1518_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys),Nb) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_1519_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Nb) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_1520_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_1521_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A4: set_nat,B4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A4),Z)
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( ( A4 = B4 )
           => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A4),Xb),X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_1522_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_1523_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1524_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1525_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_1526_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_1527_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_1528_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_1529_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_1530_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_1531_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg1Leaf
tff(fact_1532_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ).

% deg_1_Leaf
tff(fact_1533_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A3: $o,B3: $o] : Ta = vEBT_Leaf((A3),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_1534_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_1535_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_13)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X) ) ) ) ).

% succ_none_empty
tff(fact_1536_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_1537_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I,Y) = list_u1324408373059187874T_VEBT(Xs,I,Y) ).

% list_update_overwrite
tff(fact_1538_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1539_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_1540_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1541_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_1542_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1543_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_1544_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_1545_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1546_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_1547_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1548_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_1549_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1550_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_1551_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1552_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1553_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_1554_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1555_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1556_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1557_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1558_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1559_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1560_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1561_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1562_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1563_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1564_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1565_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_1566_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_1567_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1568_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_1569_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_1570_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_1571_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1572_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1573_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_1574_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1575_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1576_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_1577_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1578_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_1579_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1580_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1581_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_1582_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1583_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_1584_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1585_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1586_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_1587_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1588_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_1589_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1590_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1591_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_1592_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1593_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_1594_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_1595_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_1596_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_1597_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_1598_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_1599_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_1600_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1601_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1602_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1603_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1604_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1605_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_1606_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_1607_diff__zero,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_zero
tff(fact_1608_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_1609_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_1610_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_1611_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_1612_diff__0__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_0_right
tff(fact_1613_diff__0__right,axiom,
    ! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).

% diff_0_right
tff(fact_1614_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_1615_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_1616_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_1617_diff__self,axiom,
    ! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).

% diff_self
tff(fact_1618_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_1619_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1620_div__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_1621_div__0,axiom,
    ! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).

% div_0
tff(fact_1622_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_1623_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1624_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1625_div__by__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1626_div__by__0,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1627_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_1628_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1629_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_1630_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1631_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_1632_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1633_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_1634_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1635_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_1636_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1637_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1638_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( divide_divide_rat(C,A) = divide_divide_rat(C,B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1639_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1640_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1641_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1642_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1643_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_1644_division__ring__divide__zero,axiom,
    ! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_1645_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_1646_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_1647_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_1648_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_1649_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_1650_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1651_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_1652_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1653_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_1654_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_1655_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_1656_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_1657_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_1658_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_1659_List_Ofinite__set,axiom,
    ! [Xs: list_int] : finite_finite_int(set_int2(Xs)) ).

% List.finite_set
tff(fact_1660_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_1661_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : finite3207457112153483333omplex(set_complex2(Xs)) ).

% List.finite_set
tff(fact_1662_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1663_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).

% Nat.add_0_right
tff(fact_1664_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1665_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1666_mult__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1667_mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1668_mult__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1669_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1670_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_1671_length__list__update,axiom,
    ! [Xs: list_o,I: nat,Xb: $o] : size_size_list_o(list_update_o(Xs,I,(Xb))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_1672_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] : size_size_list_nat(list_update_nat(Xs,I,Xb)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_1673_length__list__update,axiom,
    ! [Xs: list_int,I: nat,Xb: int] : size_size_list_int(list_update_int(Xs,I,Xb)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_1674_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_1675_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_1676_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_1677_max__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).

% max_nat.right_neutral
tff(fact_1678_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1679_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).

% max_nat.left_neutral
tff(fact_1680_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1681_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_1682_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_1683_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_1684_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Xb: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_1685_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,Xb: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_1686_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_1687_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1688_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_1689_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_1690_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_1691_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_1692_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_1693_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_1694_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_1695_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_1696_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_1697_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_1698_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_1699_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_1700_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_1701_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_1702_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_1703_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_1704_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1705_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1706_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1707_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1708_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1709_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1710_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1711_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1712_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1713_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1714_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1715_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1716_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1717_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1718_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1719_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_1720_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_1721_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_1722_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_1723_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_1724_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_1725_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_1726_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_1727_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_1728_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_1729_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_1730_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_1731_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_1732_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_1733_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_1734_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_1735_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1736_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1737_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1738_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1739_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1740_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1741_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1742_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_1743_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1744_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1745_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1746_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1747_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1748_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1749_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1750_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_1751_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_1752_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1753_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1754_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_1755_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_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_1756_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_1757_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_1758_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_1759_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1760_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_1761_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_1762_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1763_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1764_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1765_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1766_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1767_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1768_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1769_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1770_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1771_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1772_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_1773_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(divide_divide_int(A),B)) ).

% div_mult_mult1_if
tff(fact_1774_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_1775_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult2
tff(fact_1776_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_1777_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ).

% div_mult_mult1
tff(fact_1778_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1779_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1780_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,divide_divide_rat(A,B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1781_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1782_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1783_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1784_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1785_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1786_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1787_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1788_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1789_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1790_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1791_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1792_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1793_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_1794_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% div_self
tff(fact_1795_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% div_self
tff(fact_1796_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_1797_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_1798_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1799_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1800_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1801_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1802_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1803_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = divide_divide_rat(A,B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1804_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_1805_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).

% divide_self
tff(fact_1806_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,A) = one_one_rat ) ) ).

% divide_self
tff(fact_1807_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_1808_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_1809_divide__self__if,axiom,
    ! [A: rat] :
      divide_divide_rat(A,A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_1810_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_1811_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( divide_divide_rat(B,A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_1812_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_1813_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = divide_divide_rat(B,A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_1814_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_1815_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat(one_one_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_1816_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_1817_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = divide_divide_rat(one_one_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_1818_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_1819_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_1820_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_1821_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_1822_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_1823_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_1824_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_1825_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_real ).

% power_zero_numeral
tff(fact_1826_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_int ).

% power_zero_numeral
tff(fact_1827_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_1828_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_1829_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_1830_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_1831_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_1832_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_1833_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_1834_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_1835_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_1836_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_1837_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_1838_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_1839_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_1840_power__Suc0__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1841_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1842_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1843_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1844_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1845_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_1846_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(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_1847_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)),numeral_numeral_real(V),numeral_numeral_real(U)) ).

% max_number_of(1)
tff(fact_1848_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)),numeral_numeral_rat(V),numeral_numeral_rat(U)) ).

% max_number_of(1)
tff(fact_1849_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_1850_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_1851_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(3)
tff(fact_1852_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(3)
tff(fact_1853_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_max_rat(zero_zero_rat,numeral_numeral_rat(Xb)) = numeral_numeral_rat(Xb) ).

% max_0_1(3)
tff(fact_1854_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_1855_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_1856_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(4)
tff(fact_1857_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(Xb)),zero_zero_real) = numeral_numeral_real(Xb) ).

% max_0_1(4)
tff(fact_1858_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_max_rat(numeral_numeral_rat(Xb),zero_zero_rat) = numeral_numeral_rat(Xb) ).

% max_0_1(4)
tff(fact_1859_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_1860_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_1861_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_1862_max__0__1_I1_J,axiom,
    ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_1863_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_1864_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_1865_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_1866_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_1867_max__0__1_I2_J,axiom,
    ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_1868_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_1869_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_1870_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_1871_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_1872_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(5)
tff(fact_1873_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(5)
tff(fact_1874_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(Xb)) = numeral_numeral_rat(Xb) ).

% max_0_1(5)
tff(fact_1875_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_1876_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_1877_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(6)
tff(fact_1878_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(Xb)),one_one_real) = numeral_numeral_real(Xb) ).

% max_0_1(6)
tff(fact_1879_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_max_rat(numeral_numeral_rat(Xb),one_one_rat) = numeral_numeral_rat(Xb) ).

% max_0_1(6)
tff(fact_1880_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_1881_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_1882_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1883_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1884_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_1885_mult__less__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_1886_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_1887_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1888_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).

% div_by_Suc_0
tff(fact_1889_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1890_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_1891_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_1892_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_1893_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1894_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_1895_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1896_mod__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_1897_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) = $ite(K2 = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_1898_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_1899_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_1900_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_1901_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I)
     => ( list_update_int(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_1902_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_1903_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_1904_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_1905_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_1906_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_1907_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_1908_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_1909_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_1910_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_1911_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_1912_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_1913_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_1914_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_1915_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_1916_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_1917_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_1918_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_1919_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1920_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1921_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1922_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1923_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1924_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1925_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_1926_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_1927_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_1928_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_1929_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1930_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) = A )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1931_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1932_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1933_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1934_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1935_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1936_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1937_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( divide_divide_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1938_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1939_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1940_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(B,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1941_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_1942_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self1
tff(fact_1943_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_1944_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self2
tff(fact_1945_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_1946_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self3
tff(fact_1947_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_1948_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_mult_self4
tff(fact_1949_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1950_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1951_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1952_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(power_power_int(A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1953_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1954_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_1955_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_1956_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1957_mult__le__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_1958_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_1959_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_1960_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_1961_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_1962_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_1963_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_1964_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_1965_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_1966_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_1967_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_1968_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_1969_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_1970_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_1971_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_1972_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1973_power__strict__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1974_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1975_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1976_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1977_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1978_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1979_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1980_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_1981_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1982_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1983_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1984_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1985_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_1986_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_1987_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_1988_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_1989_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_1990_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_1991_one__div__two__eq__zero,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_1992_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_1993_bits__1__div__2,axiom,
    aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_1994_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1995_power__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1996_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1997_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1998_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1999_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_2000_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_2001_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_2002_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_2003_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_2004_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_2005_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_2006_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_2007_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_2008_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_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_2009_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_2010_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_2011_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2012_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2013_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2014_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2015_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2016_add__self__mod__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_2017_mod2__gr__0,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_2018_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2019_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2020_zero__reorient,axiom,
    ! [Xb: complex] :
      ( ( zero_zero_complex = Xb )
    <=> ( Xb = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_2021_zero__reorient,axiom,
    ! [Xb: real] :
      ( ( zero_zero_real = Xb )
    <=> ( Xb = zero_zero_real ) ) ).

% zero_reorient
tff(fact_2022_zero__reorient,axiom,
    ! [Xb: rat] :
      ( ( zero_zero_rat = Xb )
    <=> ( Xb = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_2023_zero__reorient,axiom,
    ! [Xb: nat] :
      ( ( zero_zero_nat = Xb )
    <=> ( Xb = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_2024_zero__reorient,axiom,
    ! [Xb: int] :
      ( ( zero_zero_int = Xb )
    <=> ( Xb = zero_zero_int ) ) ).

% zero_reorient
tff(fact_2025_list__update__swap,axiom,
    ! [I: nat,I5: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( I != I5 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I5,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I5,X5),I,Xb) ) ) ).

% list_update_swap
tff(fact_2026_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2027_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_2028_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X4)
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2029_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_2030_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_2031_max__add__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,ord_max_rat(Xb,Y)),Z) = ord_max_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Z),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).

% max_add_distrib_left
tff(fact_2032_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_2033_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_2034_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_2035_max__add__distrib__right,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),ord_max_rat(Y,Z)) = ord_max_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2036_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_2037_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_2038_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_2039_max__diff__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : minus_minus_rat(ord_max_rat(Xb,Y),Z) = ord_max_rat(minus_minus_rat(Xb,Z),minus_minus_rat(Y,Z)) ).

% max_diff_distrib_left
tff(fact_2040_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_2041_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_2042_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_2043_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_2044_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_2045_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_2046_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_2047_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2048_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_2049_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_2050_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2051_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q3)) ).

% nat_add_max_left
tff(fact_2052_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ).

% nat_add_max_right
tff(fact_2053_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_2054_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).

% nat_mult_max_right
tff(fact_2055_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_max_left
tff(fact_2056_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,D2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_2057_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_2058_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_2059_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_2060_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_2061_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2062_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A4)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A4 ) ).

% finite_list
tff(fact_2063_finite__list,axiom,
    ! [A4: set_int] :
      ( finite_finite_int(A4)
     => ? [Xs2: list_int] : set_int2(Xs2) = A4 ) ).

% finite_list
tff(fact_2064_finite__list,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A4 ) ).

% finite_list
tff(fact_2065_finite__list,axiom,
    ! [A4: set_complex] :
      ( finite3207457112153483333omplex(A4)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A4 ) ).

% finite_list
tff(fact_2066_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_2067_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_2068_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_2069_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_2070_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_2071_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_2072_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_2073_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_2074_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_2075_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_2076_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_2077_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_2078_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_2079_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: 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),D22)
       => ? [E2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2080_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: 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),D22)
       => ? [E2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2081_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_2082_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != numeral_numeral_real(Nb) ).

% zero_neq_numeral
tff(fact_2083_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != numeral_numeral_rat(Nb) ).

% zero_neq_numeral
tff(fact_2084_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_2085_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_2086_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_2087_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2088_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2089_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2090_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2091_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_2092_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2093_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2094_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2095_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2096_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_2097_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2098_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2099_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2100_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2101_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2102_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2103_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2104_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2105_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2106_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2107_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2108_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2109_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2110_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2111_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2112_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2113_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2114_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2115_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2116_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2117_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2118_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2119_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2120_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2121_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_2122_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_2123_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_2124_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_2125_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_2126_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_2127_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_2128_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_2129_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_2130_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_2131_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_2132_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_2133_verit__sum__simplify,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% verit_sum_simplify
tff(fact_2134_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_2135_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_2136_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_2137_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2138_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_2139_power__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2140_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2141_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2142_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2143_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2144_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_2145_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_2146_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_2147_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2148_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_2149_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2150_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_2151_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X4),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X4),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_2152_zero__induct,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [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_2153_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_2154_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_2155_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_2156_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_2157_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va3: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va3)) ) ) ).

% vebt_buildup.cases
tff(fact_2158_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N: nat] : Xb != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_2159_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu2),(Uv2))),Uw2) ).

% VEBT_internal.membermima.simps(1)
tff(fact_2160_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_2161_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_2162_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_2163_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_2164_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_2165_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_2166_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_2167_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_2168_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_2169_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_2170_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_2171_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_2172_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_2173_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_2174_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).

% minus_nat.diff_0
tff(fact_2175_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
    <=> ( ( K2 = zero_zero_nat )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2176_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_2177_max__def__raw,axiom,
    ! [X: extended_enat,Xa: extended_enat] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Xa) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(X),Xa),Xa,X) ).

% max_def_raw
tff(fact_2178_max__def__raw,axiom,
    ! [X: set_nat,Xa: set_nat] :
      ord_max_set_nat(X,Xa) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X),Xa),Xa,X) ).

% max_def_raw
tff(fact_2179_max__def__raw,axiom,
    ! [X: rat,Xa: rat] :
      ord_max_rat(X,Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_2180_max__def__raw,axiom,
    ! [X: num,Xa: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_2181_max__def__raw,axiom,
    ! [X: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_2182_max__def__raw,axiom,
    ! [X: int,Xa: int] :
      aa_int_int(ord_max_int(X),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_2183_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2184_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2185_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2186_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2187_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2188_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2189_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2190_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2191_lambda__zero,axiom,
    aTP_Lamm_be = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_2192_lambda__zero,axiom,
    aTP_Lamm_bf = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_2193_lambda__zero,axiom,
    aTP_Lamm_bg = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_2194_lambda__zero,axiom,
    aTP_Lamm_bh = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_2195_lambda__zero,axiom,
    aTP_Lamm_bi = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_2196_finite__lists__length__eq,axiom,
    ! [A4: set_complex,Nb: nat] :
      ( finite3207457112153483333omplex(A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bj(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_2197_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bk(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_2198_finite__lists__length__eq,axiom,
    ! [A4: set_o,Nb: nat] :
      ( finite_finite_o(A4)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_bl(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_2199_finite__lists__length__eq,axiom,
    ! [A4: set_int,Nb: nat] :
      ( finite_finite_int(A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bm(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_2200_finite__lists__length__eq,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bn(A4),Nb))) ) ).

% finite_lists_length_eq
tff(fact_2201_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_2202_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2203_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2204_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2205_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2206_set__update__subsetI,axiom,
    ! [Xs: list_real,A4: set_real,Xb: real,I: nat] :
      ( ord_less_eq_set_real(set_real2(Xs),A4)
     => ( member_real(Xb,A4)
       => ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_2207_set__update__subsetI,axiom,
    ! [Xs: list_complex,A4: set_complex,Xb: complex,I: nat] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),A4)
     => ( member_complex(Xb,A4)
       => ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_2208_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat,Xb: product_prod_nat_nat,I: nat] :
      ( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs),A4)
     => ( member8440522571783428010at_nat(Xb,A4)
       => ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_2209_set__update__subsetI,axiom,
    ! [Xs: list_int,A4: set_int,Xb: int,I: nat] :
      ( ord_less_eq_set_int(set_int2(Xs),A4)
     => ( member_int(Xb,A4)
       => ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_2210_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,Xb: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A4)
     => ( member_VEBT_VEBT(Xb,A4)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb)),A4) ) ) ).

% set_update_subsetI
tff(fact_2211_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,Xb: nat,I: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A4)
     => ( aa_set_nat_o(member_nat(Xb),A4)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xb))),A4) ) ) ).

% set_update_subsetI
tff(fact_2212_finite__lists__length__le,axiom,
    ! [A4: set_complex,Nb: nat] :
      ( finite3207457112153483333omplex(A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bo(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_2213_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bp(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_2214_finite__lists__length__le,axiom,
    ! [A4: set_o,Nb: nat] :
      ( finite_finite_o(A4)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_bq(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_2215_finite__lists__length__le,axiom,
    ! [A4: set_int,Nb: nat] :
      ( finite_finite_int(A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_br(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_2216_finite__lists__length__le,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bs(A4),Nb))) ) ).

% finite_lists_length_le
tff(fact_2217_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_le_numeral
tff(fact_2218_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(Nb)) ).

% zero_le_numeral
tff(fact_2219_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_2220_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_2221_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_2222_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_2223_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_2224_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_2225_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2226_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2227_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2228_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2229_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_2230_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_2231_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_2232_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2233_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2234_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2235_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2236_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2237_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2238_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2239_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_2240_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2241_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2242_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2243_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_2244_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2245_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2246_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2247_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2248_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2249_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2250_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_2251_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_2252_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2253_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2254_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_2255_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono
tff(fact_2256_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono
tff(fact_2257_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_2258_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_2259_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2260_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2261_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2262_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono
tff(fact_2263_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono
tff(fact_2264_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_2265_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_2266_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2267_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2268_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2269_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2270_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2271_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_2272_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_2273_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% split_mult_pos_le
tff(fact_2274_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_2275_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_2276_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_2277_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_2278_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2279_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2280_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2281_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2282_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2283_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2284_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2285_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_2286_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_2287_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_2288_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_2289_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_2290_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(Nb)) ).

% zero_less_numeral
tff(fact_2291_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(Nb)) ).

% zero_less_numeral
tff(fact_2292_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_2293_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_2294_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_2295_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_2296_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_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2307_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2308_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2309_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2310_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_2311_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_2312_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_2313_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_2314_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2315_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2316_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2317_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2318_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_2319_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_2320_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_2321_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_2322_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2323_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2324_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2325_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2326_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_2327_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_2328_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_2329_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_2330_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_2331_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(aa_rat_fun_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_2332_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_2333_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_2334_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_2335_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(aa_rat_fun_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_2336_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_2337_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_2338_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu2),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_2339_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2340_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2341_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2342_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2343_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2344_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2345_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2346_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2347_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2348_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2349_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2350_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2351_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2352_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2353_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2354_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2355_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2356_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2357_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2358_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2359_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_2360_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2361_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2362_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2363_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2364_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2365_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2366_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2367_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2368_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2369_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2370_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2371_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2372_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_2373_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_2374_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2375_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_2376_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_2377_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_2378_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2379_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_2380_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2381_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2382_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_2383_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_2384_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2385_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2386_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2387_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_2388_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2389_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2390_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_2391_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_2392_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_2393_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_2394_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_2395_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_2396_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2397_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2398_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_2399_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_2400_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_2401_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_2402_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2403_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2404_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_2405_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_2406_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_2407_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_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_2420_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_2421_add__less__zeroD,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,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_2422_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_2423_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_2424_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_2425_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_2426_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_2427_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C2: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2428_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_2429_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_pos
tff(fact_2430_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_2431_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_2432_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_2433_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_2434_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_2435_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_2436_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
        (A),
        some_nat(zero_zero_nat),
        $ite((B),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_2437_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2438_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2439_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_right_mono
tff(fact_2440_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_right_mono
tff(fact_2441_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2442_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_2443_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_2444_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),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2445_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_2446_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,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2447_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_2448_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,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2449_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_2450_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),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2451_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2452_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),divide_divide_rat(A,C)) ) ) ).

% divide_right_mono_neg
tff(fact_2453_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_2454_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_2455_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_2456_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2457_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2458_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2459_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)) ) ) ).

% divide_strict_right_mono
tff(fact_2460_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_2461_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_2462_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2463_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2464_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2465_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A,B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2466_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_2467_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),divide_divide_rat(Xb,Y)) ) ) ).

% divide_pos_pos
tff(fact_2468_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_2469_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,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_2470_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_2471_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,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_2472_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_2473_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),divide_divide_rat(Xb,Y)) ) ) ).

% divide_neg_neg
tff(fact_2474_zero__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_power
tff(fact_2475_zero__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_power
tff(fact_2476_zero__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_le_power
tff(fact_2477_zero__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_power
tff(fact_2478_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_2479_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_2480_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_2481_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono
tff(fact_2482_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_2483_zero__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_less_power
tff(fact_2484_zero__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_less_power
tff(fact_2485_zero__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_less_power
tff(fact_2486_zero__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_less_power
tff(fact_2487_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xb),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2488_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: 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,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2489_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(Xb,Y) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2490_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_2491_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2492_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( divide_divide_rat(B,C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2493_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_2494_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2495_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2496_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2497_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2498_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
       => ( divide_divide_rat(B,C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2499_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2500_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2501_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
       => ( A = divide_divide_rat(B,C) ) ) ) ).

% eq_divide_imp
tff(fact_2502_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2503_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
      <=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2504_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B,C) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2505_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2506_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2507_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = divide_divide_rat(B,C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2508_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2509_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2510_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat(A,B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2511_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2512_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2513_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2514_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2515_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2516_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_2517_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2518_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2519_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2520_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2521_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = A )
    <=> ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2522_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
          & 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),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_2523_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M5: nat] : Nb = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_2524_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Nb))
         => 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),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_2525_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_2526_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2527_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_2528_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_2529_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K) ) ) ) ).

% ex_least_nat_le
tff(fact_2530_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_2531_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_2532_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_2533_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2534_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2535_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2536_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2537_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_2538_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ) ).

% mult_less_mono1
tff(fact_2539_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K2: 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),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ) ).

% mult_less_mono2
tff(fact_2540_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2541_nat__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_2542_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2543_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2544_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_2545_nat__power__less__imp__less,axiom,
    ! [I: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_2546_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2547_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_2548_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_2549_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q2: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).

% mod_eq_0D
tff(fact_2550_vebt__insert_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X4)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S),X4)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S),X4)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),X4)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: 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,Va3)),TreeList,Summary),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_2551_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Uw)
     => ( ! [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,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),X4)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),X4)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_2552_vebt__succ_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B3: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat)
     => ( ! [Uv: $o,Uw: $o,N: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va2)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve)
           => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi)
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: 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,Va3)),TreeList,Summary),X4) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_2553_vebt__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X4)
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),X4)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X4)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X4)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X4: 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,Va3)),TreeList,Summary),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_2554_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts2,S2),Xb) = vEBT_Node(Info,zero_zero_nat,Ts2,S2) ).

% vebt_insert.simps(2)
tff(fact_2555_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_2556_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : Xb != vEBT_Leaf((Uu),$true)
         => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_2557_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(Xb,set_real2(list_update_real(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_2558_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(Xb,set_complex2(list_update_complex(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_2559_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_P6011104703257516679at_nat,Xb: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s5460976970255530739at_nat(Xs))
     => member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_2560_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(Xb,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_2561_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o((Xb),set_o2(list_update_o(Xs,Nb,(Xb)))) ) ).

% set_update_memI
tff(fact_2562_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_2563_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(Xb,set_int2(list_update_int(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_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_2572_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_2573_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : Xb != vEBT_Leaf((Uu),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2574_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2575_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2576_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2577_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2578_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2579_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2580_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2581_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2582_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2583_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2584_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2585_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2586_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2587_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2588_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2589_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2590_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2591_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2592_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2593_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2594_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2595_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2596_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2597_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2598_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2599_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2600_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2601_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2602_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2603_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2604_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2605_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2606_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2607_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2608_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2609_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2610_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2611_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2612_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2613_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2614_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2615_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2616_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2617_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2618_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2619_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2620_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2621_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2622_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2623_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2624_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2625_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2626_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2627_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2628_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2629_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2630_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2631_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2632_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2633_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2634_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2635_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2636_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2637_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2638_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2639_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2640_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2641_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2642_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2643_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2644_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2645_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2646_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2647_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2648_field__le__epsilon,axiom,
    ! [Xb: real,Y: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2649_field__le__epsilon,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2650_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2651_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2652_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_2653_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)) ) ) ).

% div_positive
tff(fact_2654_frac__le,axiom,
    ! [Y: real,Xb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_2655_frac__le,axiom,
    ! [Y: rat,Xb: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_le
tff(fact_2656_frac__less,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_2657_frac__less,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less
tff(fact_2658_frac__less2,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_2659_frac__less2,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less2
tff(fact_2660_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2661_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2662_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_2663_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,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_2664_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_2665_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),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2666_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_2667_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),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2668_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_2669_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,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_2670_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2671_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2672_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2673_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2674_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2675_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2676_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_2677_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_2678_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_2679_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(aa_rat_fun_rat_rat(times_times_rat,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2680_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_2681_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_2682_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(aa_rat_fun_rat_rat(times_times_rat,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2683_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_2684_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_2685_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2686_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_2687_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_2688_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_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_2689_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_2690_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2691_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2692_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2693_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2694_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_2695_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2696_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_2697_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_2698_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2699_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_2700_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2701_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2702_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_2703_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_2704_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_2705_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_2706_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2707_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2708_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2709_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2710_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_2711_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(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),divide_divide_rat(Xb,Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2712_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_2713_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(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2714_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2715_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2716_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2717_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2718_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2719_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2720_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2721_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2722_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_2723_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2724_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_2725_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_2726_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2727_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2728_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2729_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2730_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2731_power__le__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_2732_power__le__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2733_power__le__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2734_power__le__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_2735_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2736_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = numeral_numeral_real(W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C),numeral_numeral_real(W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2737_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2738_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2739_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( numeral_numeral_real(W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) = B,numeral_numeral_real(W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2740_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2741_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2742_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2743_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2744_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2745_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2746_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2747_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2748_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2749_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2750_add__frac__num,axiom,
    ! [Y: complex,Xb: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2751_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_2752_add__frac__num,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(Xb,Y)),Z) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_2753_add__num__frac,axiom,
    ! [Y: complex,Z: complex,Xb: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2754_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_2755_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(Xb,Y)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_2756_add__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2757_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_2758_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),divide_divide_rat(Y,Z)) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z)),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2759_divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2760_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_2761_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,divide_divide_rat(Xb,Z)),Y) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2762_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2763_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2764_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2765_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_2766_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2767_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2768_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2769_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2770_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2771_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2772_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2773_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2774_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2775_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2776_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      minus_minus_rat(A,divide_divide_rat(B,Z)) = $ite(Z = zero_zero_rat,A,divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z),B),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2777_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2778_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2779_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(Xb,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2780_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2781_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_2782_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(Xb,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2783_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2784_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_2785_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(Xb,Z),Y) = divide_divide_rat(minus_minus_rat(Xb,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2786_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2787_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2788_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2789_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2790_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2791_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2792_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_2793_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_2794_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_2795_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_2796_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_2797_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_2798_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_2799_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_2800_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_2801_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_2802_length__pos__if__in__set,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2803_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_2804_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_2805_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_2806_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_2807_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( member_int(Xb,set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2808_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_2809_nat__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_2810_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_2811_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split
tff(fact_2812_power__gt__expt,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K2)) ) ).

% power_gt_expt
tff(fact_2813_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Mb)) ) ) ).

% div_le_mono2
tff(fact_2814_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_2815_nat__one__le__power,axiom,
    ! [I: nat,Nb: 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),Nb)) ) ).

% nat_one_le_power
tff(fact_2816_div__less__iff__less__mult,axiom,
    ! [Q3: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Q3)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_2817_nat__mult__div__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_2818_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Mb )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2819_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_2820_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_2821_div__less__mono,axiom,
    ! [A4: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_2822_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) ).

% vebt_insert.simps(3)
tff(fact_2823_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_2824_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_2825_vebt__mint_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] : Xb != vEBT_Leaf((A3),(B3))
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
       => ~ ! [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_2826_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) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
           => (Y) )
         => ( ( ? [Uu: $o] : Xb = vEBT_Leaf((Uu),$true)
             => (Y) )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2827_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2828_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2829_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2830_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2831_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2832_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2833_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2834_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2835_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2836_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2837_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2838_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2839_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2840_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2841_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2842_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2843_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2844_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2845_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2846_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2847_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2848_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2849_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2850_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2851_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_2852_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(aa_rat_fun_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_2853_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_2854_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_2855_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_2856_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2857_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2858_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono
tff(fact_2859_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2860_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2861_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2862_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2863_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2864_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2865_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2866_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2867_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_2868_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(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2869_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_2870_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(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),divide_divide_rat(Xb,Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2871_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2872_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2873_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2874_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2875_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2876_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2877_convex__bound__le,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2878_convex__bound__le,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2879_convex__bound__le,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2880_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2881_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2882_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2883_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2884_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_2885_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_2886_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_2887_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_2888_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2889_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2890_power__Suc__less,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2891_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2892_power__Suc__le__self,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2893_power__Suc__le__self,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2894_power__Suc__le__self,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2895_power__Suc__le__self,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_2896_power__Suc__less__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_2897_power__Suc__less__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_2898_power__Suc__less__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_2899_power__Suc__less__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_2900_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2901_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2902_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2903_power__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2904_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2905_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2906_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2907_power__decreasing,axiom,
    ! [Nb: nat,N3: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2908_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_2909_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_2910_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_2911_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_2912_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_2913_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (A3),
                  some_nat(zero_zero_nat),
                  $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( 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_2914_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (B3),
                  some_nat(one_one_nat),
                  $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( 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_2915_self__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% self_le_power
tff(fact_2916_self__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_2917_self__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% self_le_power
tff(fact_2918_self__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% self_le_power
tff(fact_2919_one__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% one_less_power
tff(fact_2920_one__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_2921_one__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_2922_one__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% one_less_power
tff(fact_2923_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) ) )
       => ( ! [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,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) )
                   => ( 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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_2924_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_2925_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_2926_power__diff,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_2927_power__diff,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_2928_power__diff,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),Mb),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_2929_power__diff,axiom,
    ! [A: nat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_2930_power__diff,axiom,
    ! [A: int,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_2931_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_2932_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_2933_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_2934_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_2935_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_2936_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2937_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I3)),J3) )
               => aa_nat_o(P,I3) ) ) ) ) ) ).

% split_div
tff(fact_2938_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_2939_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_2940_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_2941_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I3)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2942_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_2943_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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(Mia),Maa)),zero_zero_nat,Va,Vb)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_2944_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve2) = none_nat ).

% vebt_succ.simps(4)
tff(fact_2945_convex__bound__lt,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2946_convex__bound__lt,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2947_convex__bound__lt,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2948_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2949_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2950_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2951_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2952_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2953_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2954_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_2955_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_2956_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(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))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_2957_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),minus_minus_rat(V,U)),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_2958_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2959_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2960_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2961_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_2962_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_2963_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_2964_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_2965_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_2966_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_2967_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_2968_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_2969_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_2970_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_2971_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_2972_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2973_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2974_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2975_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2976_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2977_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2978_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2979_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2980_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_2981_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_2982_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_2983_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_2984_power__eq__if,axiom,
    ! [P2: complex,Mb: nat] :
      aa_nat_complex(power_power_complex(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P2),aa_nat_complex(power_power_complex(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_2985_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_2986_power__eq__if,axiom,
    ! [P2: rat,Mb: nat] :
      aa_nat_rat(power_power_rat(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_2987_power__eq__if,axiom,
    ! [P2: nat,Mb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_2988_power__eq__if,axiom,
    ! [P2: int,Mb: nat] :
      aa_nat_int(power_power_int(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(power_power_int(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_2989_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_2990_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% power_minus_mult
tff(fact_2991_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_minus_mult
tff(fact_2992_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_2993_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_2994_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_2995_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_2996_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_2997_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_2998_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2) = none_nat ).

% vebt_succ.simps(5)
tff(fact_2999_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_3000_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_3001_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_3002_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_3003_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_3004_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(aa_rat_fun_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_3005_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_3006_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_3007_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(aa_rat_fun_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_3008_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_3009_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_3010_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(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_3011_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_3012_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_3013_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(aa_rat_fun_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_3014_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_3015_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3016_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3017_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_3018_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_3019_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3020_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3021_zero__le__even__power_H,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3022_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_3023_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_3024_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_3025_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_3026_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3027_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3028_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3029_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3030_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3031_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3032_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3033_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3034_odd__power__less__zero,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_3035_odd__power__less__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_3036_odd__power__less__zero,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_3037_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_3038_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3039_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_3040_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3041_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Uu: $o,Uv: $o] : Xb != vEBT_Leaf((Uu),(Uv))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3042_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( (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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3043_mod__double__modulus,axiom,
    ! [Mb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3044_mod__double__modulus,axiom,
    ! [Mb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( ( aa_int_int(modulo_modulo_int(Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(modulo_modulo_int(Xb),Mb) )
          | ( aa_int_int(modulo_modulo_int(Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3045_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3046_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3047_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => (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,Va3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary)
                   => ( (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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3048_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
         => ( ! [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,Va3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3049_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)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3050_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(aa_rat_fun_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),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3051_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A3: $o,B3: $o] : A1 = vEBT_Leaf((A3),(B3))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                     => ( ( M = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                         => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                       => ( ( M = aa_nat_nat(suc,N) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                           => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                         => ( ( M = N )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                             => ( ! [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))),M))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [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))),M))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                           => ( ( M = aa_nat_nat(suc,N) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                               => ( ! [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))),M))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [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))),M))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3052_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [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,Ma2),N4) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [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,Ma2),N4) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3053_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3054_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3055_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_3056_finite__Collect__le__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bt,K2))) ).

% finite_Collect_le_nat
tff(fact_3057_finite__Collect__less__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu,K2))) ).

% finite_Collect_less_nat
tff(fact_3058_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_real_fun_real_o] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(aa_fun8588574755773516687real_o(aTP_Lamm_bv(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => finite_finite_real(collect_real(aa_real_fun_real_o(aTP_Lamm_bw(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3059_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_nat_fun_real_o] :
      ( finite_finite_real(collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1522457866930155023_nat_o(aTP_Lamm_bx(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => aa_set_nat_o(finite_finite_nat,collect_nat(aa_real_fun_nat_o(aTP_Lamm_by(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3060_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_co4998825143550818160real_o] :
      ( finite_finite_real(collect_real(P))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun4501470310227681423plex_o(aTP_Lamm_bz(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => finite3207457112153483333omplex(collect_complex(aa_rea2382570592068293209plex_o(aTP_Lamm_ca(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3061_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_o,Q: fun_real_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( finite_finite_real(collect_real(aa_fun8982401453414893235real_o(aTP_Lamm_cb(P),Q)))
      <=> ! [Y5: nat] :
            ( aa_nat_o(P,Y5)
           => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_cc(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3062_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_o,Q: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6854469565811772723_nat_o(aTP_Lamm_cd(P),Q)))
      <=> ! [Y5: nat] :
            ( aa_nat_o(P,Y5)
           => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ce(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3063_finite__Collect__bounded__ex,axiom,
    ! [P: fun_nat_o,Q: fun_co5676208753217392844_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun3010802591474140851plex_o(aTP_Lamm_cf(P),Q)))
      <=> ! [Y5: nat] :
            ( aa_nat_o(P,Y5)
           => finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cg(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3064_finite__Collect__bounded__ex,axiom,
    ! [P: fun_complex_o,Q: fun_re7319081385389557872plex_o] :
      ( finite3207457112153483333omplex(collect_complex(P))
     => ( finite_finite_real(collect_real(aa_fun7329550764591405201real_o(aTP_Lamm_ch(P),Q)))
      <=> ! [Y5: complex] :
            ( aa_complex_o(P,Y5)
           => finite_finite_real(collect_real(aa_com1951324360330529881real_o(aTP_Lamm_ci(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3065_finite__Collect__bounded__ex,axiom,
    ! [P: fun_complex_o,Q: fun_na1554462459066888140plex_o] :
      ( finite3207457112153483333omplex(collect_complex(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5940009423621039633_nat_o(aTP_Lamm_cj(P),Q)))
      <=> ! [Y5: complex] :
            ( aa_complex_o(P,Y5)
           => aa_set_nat_o(finite_finite_nat,collect_nat(aa_complex_fun_nat_o(aTP_Lamm_ck(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3066_finite__Collect__bounded__ex,axiom,
    ! [P: fun_complex_o,Q: fun_co1148497455491383662plex_o] :
      ( finite3207457112153483333omplex(collect_complex(P))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun7218396518982330257plex_o(aTP_Lamm_cl(P),Q)))
      <=> ! [Y5: complex] :
            ( aa_complex_o(P,Y5)
           => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_cm(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3067_finite__Collect__bounded__ex,axiom,
    ! [P: fun_real_o,Q: fun_li7444029373303974462real_o] :
      ( finite_finite_real(collect_real(P))
     => ( finite8100373058378681591st_nat(collect_list_nat(aa_fun5611870488384741519_nat_o(aTP_Lamm_cn(P),Q)))
      <=> ! [Y5: real] :
            ( aa_real_o(P,Y5)
           => finite8100373058378681591st_nat(collect_list_nat(aa_rea6225427375947155751_nat_o(aTP_Lamm_co(Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_3068_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_cp,Nb))) ) ).

% finite_roots_unity
tff(fact_3069_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cq,Nb))) ) ).

% finite_roots_unity
tff(fact_3070_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B3),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                some_nat(Mi),
                                $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,Va3))),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,Va3))),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(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_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,Va3)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3071_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb3
tff(fact_3072_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_3073_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb3
tff(fact_3074_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_3075_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_3076_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb3
tff(fact_3077_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb4
tff(fact_3078_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_3079_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb4
tff(fact_3080_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_3081_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_3082_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb4
tff(fact_3083_max__less__iff__conj,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y)),Z)
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Xb),Z)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3084_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_3085_max__less__iff__conj,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_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_3086_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_3087_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_3088_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_3089_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% set_bit_nonnegative_int_iff
tff(fact_3090_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_3091_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_3092_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_3093_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_3094_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_3095_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
        & aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A) ) ) ).

% max.bounded_iff
tff(fact_3096_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B,C)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.bounded_iff
tff(fact_3097_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.bounded_iff
tff(fact_3098_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.bounded_iff
tff(fact_3099_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.bounded_iff
tff(fact_3100_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb2
tff(fact_3101_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ord_max_rat(A,B) = B ) ) ).

% max.absorb2
tff(fact_3102_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_3103_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_3104_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb2
tff(fact_3105_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb1
tff(fact_3106_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( ord_max_rat(A,B) = A ) ) ).

% max.absorb1
tff(fact_3107_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_3108_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_3109_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb1
tff(fact_3110_finite__Diff2,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( finite3207457112153483333omplex(B4)
     => ( finite3207457112153483333omplex(minus_811609699411566653omplex(A4,B4))
      <=> finite3207457112153483333omplex(A4) ) ) ).

% finite_Diff2
tff(fact_3111_finite__Diff2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,B4))
      <=> aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_Diff2
tff(fact_3112_finite__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( finite3207457112153483333omplex(A4)
     => finite3207457112153483333omplex(minus_811609699411566653omplex(A4,B4)) ) ).

% finite_Diff
tff(fact_3113_finite__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,B4)) ) ).

% finite_Diff
tff(fact_3114_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_3115_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(divide_divide_int(K2),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_3116_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(divide_divide_int(K2),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_3117_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(modulo_modulo_int(K2),L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_3118_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(modulo_modulo_int(K2),L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_3119_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(K2),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,K2),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_3120_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K2),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),K2) ) ).

% half_nonnegative_int_iff
tff(fact_3121_set__bit__greater__eq,axiom,
    ! [K2: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(bit_se7879613467334960850it_int(Nb),K2)) ).

% set_bit_greater_eq
tff(fact_3122_unset__bit__less__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(Nb),K2)),K2) ).

% unset_bit_less_eq
tff(fact_3123_int__div__less__self,axiom,
    ! [Xb: int,K2: 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),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(divide_divide_int(Xb),K2)),Xb) ) ) ).

% int_div_less_self
tff(fact_3124_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(modulo_modulo_int(K2),L)) ) ).

% neg_mod_bound
tff(fact_3125_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(K2),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_3126_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(K2),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_3127_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K2),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3128_mod__pos__geq,axiom,
    ! [L: int,K2: 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),K2)
       => ( aa_int_int(modulo_modulo_int(K2),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_3129_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(Mb,Nb))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Mb)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_3130_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_3131_iadd__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_3132_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_3133_i0__lb,axiom,
    ! [Nb: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Nb) ).

% i0_lb
tff(fact_3134_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less
tff(fact_3135_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less
tff(fact_3136_finite__maxlen,axiom,
    ! [M6: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M6)
     => ? [N: nat] :
        ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X)),N) ) ) ).

% finite_maxlen
tff(fact_3137_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N: nat] :
        ! [X: list_o] :
          ( member_list_o(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X)),N) ) ) ).

% finite_maxlen
tff(fact_3138_finite__maxlen,axiom,
    ! [M6: set_list_nat] :
      ( finite8100373058378681591st_nat(M6)
     => ? [N: nat] :
        ! [X: list_nat] :
          ( member_list_nat(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X)),N) ) ) ).

% finite_maxlen
tff(fact_3139_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N: nat] :
        ! [X: list_int] :
          ( member_list_int(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X)),N) ) ) ).

% finite_maxlen
tff(fact_3140_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R3: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3141_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R3: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3142_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: int,Q5: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R3)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3143_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: int,Q5: int,R3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R3))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3144_q__pos__lemma,axiom,
    ! [B2: int,Q5: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R3))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_3145_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3146_div__pos__geq,axiom,
    ! [L: int,K2: 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),K2)
       => ( aa_int_int(divide_divide_int(K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_3147_zmod__eq__0__iff,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(Mb),D) = zero_zero_int )
    <=> ? [Q4: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_3148_zmod__eq__0D,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(Mb),D) = zero_zero_int )
     => ? [Q2: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q2) ) ).

% zmod_eq_0D
tff(fact_3149_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(modulo_modulo_int(K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3150_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_3151_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3152_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_3153_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K2: int] :
      ( aa_int_o(P,aa_int_int(divide_divide_int(Nb),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [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),K2)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P,I3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P,I3) ) ) ) ) ).

% split_zdiv
tff(fact_3154_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(divide_divide_int(A),B) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3155_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(divide_divide_int(A),B) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3156_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3157_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3158_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K2: int] :
      ( aa_int_o(P,aa_int_int(modulo_modulo_int(Nb),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [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),K2)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_3159_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A4),Nb)),
                $ite(aa_int_int(modulo_modulo_int(B4),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(divide_divide_int(B4),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_3160_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R4),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_3161_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X4)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),Nb) = A )
            & ! [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),Nb) = A ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3162_split__neg__lemma,axiom,
    ! [K2: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K2)),aa_int_int(modulo_modulo_int(Nb),K2))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_3163_split__pos__lemma,axiom,
    ! [K2: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(Nb),K2)),aa_int_int(modulo_modulo_int(Nb),K2))
      <=> ! [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),K2)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_3164_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3165_int__power__div__base,axiom,
    ! [Mb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K2),Mb)),K2) = aa_nat_int(power_power_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3166_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => ? [X4: real] :
            ( member_real(X4,A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),A)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3167_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A4)
            & aa_set_nat_o(ord_less_eq_set_nat(X4),A)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3168_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( finite_finite_rat(A4)
     => ( member_rat(A,A4)
       => ? [X4: rat] :
            ( member_rat(X4,A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),A)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3169_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( finite_finite_num(A4)
     => ( member_num(A,A4)
       => ? [X4: num] :
            ( member_num(X4,A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),A)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3170_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3171_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( finite_finite_int(A4)
     => ( member_int(A,A4)
       => ? [X4: int] :
            ( member_int(X4,A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),A)
            & ! [Xa: int] :
                ( member_int(Xa,A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3172_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( finite_finite_real(A4)
     => ( member_real(A,A4)
       => ? [X4: real] :
            ( member_real(X4,A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3173_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( member_set_nat(A,A4)
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A4)
            & aa_set_nat_o(ord_less_eq_set_nat(A),X4)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3174_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( finite_finite_rat(A4)
     => ( member_rat(A,A4)
       => ? [X4: rat] :
            ( member_rat(X4,A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X4)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3175_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( finite_finite_num(A4)
     => ( member_num(A,A4)
       => ? [X4: num] :
            ( member_num(X4,A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X4)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3176_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3177_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( finite_finite_int(A4)
     => ( member_int(A,A4)
       => ? [X4: int] :
            ( member_int(X4,A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X4)
            & ! [Xa: int] :
                ( member_int(Xa,A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3178_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3179_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3180_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_3181_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3182_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(D),B)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,C),D)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ) ).

% max.mono
tff(fact_3183_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(C,D)),ord_max_rat(A,B)) ) ) ).

% max.mono
tff(fact_3184_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ) ).

% max.mono
tff(fact_3185_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_3186_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(C),D)),aa_int_int(ord_max_int(A),B)) ) ) ).

% max.mono
tff(fact_3187_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
     => ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.orderE
tff(fact_3188_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = ord_max_rat(A,B) ) ) ).

% max.orderE
tff(fact_3189_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_3190_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_3191_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(ord_max_int(A),B) ) ) ).

% max.orderE
tff(fact_3192_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A) ) ).

% max.orderI
tff(fact_3193_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = ord_max_rat(A,B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_3194_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% max.orderI
tff(fact_3195_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_3196_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(ord_max_int(A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_3197_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)
     => ~ ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
         => ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A) ) ) ).

% max.boundedE
tff(fact_3198_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B,C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.boundedE
tff(fact_3199_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.boundedE
tff(fact_3200_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.boundedE
tff(fact_3201_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.boundedE
tff(fact_3202_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_3203_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B,C)),A) ) ) ).

% max.boundedI
tff(fact_3204_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A) ) ) ).

% max.boundedI
tff(fact_3205_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_3206_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C)),A) ) ) ).

% max.boundedI
tff(fact_3207_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
    <=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.order_iff
tff(fact_3208_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = ord_max_rat(A,B) ) ) ).

% max.order_iff
tff(fact_3209_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_3210_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_3211_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(ord_max_int(A),B) ) ) ).

% max.order_iff
tff(fact_3212_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ).

% max.cobounded1
tff(fact_3213_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),ord_max_rat(A,B)) ).

% max.cobounded1
tff(fact_3214_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded1
tff(fact_3215_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_3216_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(ord_max_int(A),B)) ).

% max.cobounded1
tff(fact_3217_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ).

% max.cobounded2
tff(fact_3218_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),ord_max_rat(A,B)) ).

% max.cobounded2
tff(fact_3219_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded2
tff(fact_3220_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_3221_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(ord_max_int(A),B)) ).

% max.cobounded2
tff(fact_3222_le__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y))
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Xb)
        | aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_3223_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),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_3224_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_3225_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_3226_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_3227_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),A)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3228_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( ord_max_rat(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_3229_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3230_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3231_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_3232_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3233_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( ord_max_rat(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_3234_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3235_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3236_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_3237_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.coboundedI1
tff(fact_3238_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),ord_max_rat(A,B)) ) ).

% max.coboundedI1
tff(fact_3239_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI1
tff(fact_3240_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_3241_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.coboundedI1
tff(fact_3242_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),B)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.coboundedI2
tff(fact_3243_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),ord_max_rat(A,B)) ) ).

% max.coboundedI2
tff(fact_3244_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI2
tff(fact_3245_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_3246_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.coboundedI2
tff(fact_3247_Diff__infinite__finite,axiom,
    ! [T: set_complex,S3: set_complex] :
      ( finite3207457112153483333omplex(T)
     => ( ~ finite3207457112153483333omplex(S3)
       => ~ finite3207457112153483333omplex(minus_811609699411566653omplex(S3,T)) ) ) ).

% Diff_infinite_finite
tff(fact_3248_Diff__infinite__finite,axiom,
    ! [T: set_nat,S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( ~ aa_set_nat_o(finite_finite_nat,S3)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S3,T)) ) ) ).

% Diff_infinite_finite
tff(fact_3249_less__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Xb)
        | aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_3250_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_3251_less__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),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_3252_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_3253_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_3254_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_3255_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)
     => ~ ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
         => ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A) ) ) ).

% max.strict_boundedE
tff(fact_3256_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% max.strict_boundedE
tff(fact_3257_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,ord_max_rat(B,C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_3258_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% max.strict_boundedE
tff(fact_3259_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_3260_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% max.strict_boundedE
tff(fact_3261_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B),A)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3262_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3263_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = ord_max_rat(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3264_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3265_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3266_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(ord_max_int(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_3267_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_3268_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_3269_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A,B)) ) ).

% max.strict_coboundedI1
tff(fact_3270_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_3271_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_3272_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_3273_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),B)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_3274_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_3275_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A,B)) ) ).

% max.strict_coboundedI2
tff(fact_3276_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_3277_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_3278_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_3279_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_real] :
      ( finite_finite_real(collect_real(P))
     => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_cr(P),F))) ) ).

% finite_image_set
tff(fact_3280_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_nat] :
      ( finite_finite_real(collect_real(P))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun2333486890442116906_nat_o(aTP_Lamm_cs(P),F))) ) ).

% finite_image_set
tff(fact_3281_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_complex] :
      ( finite_finite_real(collect_real(P))
     => finite3207457112153483333omplex(collect_complex(aa_fun1011824800444484462plex_o(aTP_Lamm_ct(P),F))) ) ).

% finite_image_set
tff(fact_3282_finite__image__set,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => finite_finite_real(collect_real(aa_fun6953832176698369998real_o(aTP_Lamm_cu(P),F))) ) ).

% finite_image_set
tff(fact_3283_finite__image__set,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_cv(P),F))) ) ).

% finite_image_set
tff(fact_3284_finite__image__set,axiom,
    ! [P: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => finite3207457112153483333omplex(collect_complex(aa_fun2681562978987535434plex_o(aTP_Lamm_cw(P),F))) ) ).

% finite_image_set
tff(fact_3285_finite__image__set,axiom,
    ! [P: fun_complex_o,F: fun_complex_real] :
      ( finite3207457112153483333omplex(collect_complex(P))
     => finite_finite_real(collect_real(aa_fun6876059748394487152real_o(aTP_Lamm_cx(P),F))) ) ).

% finite_image_set
tff(fact_3286_finite__image__set,axiom,
    ! [P: fun_complex_o,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(collect_complex(P))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8801047744250750888_nat_o(aTP_Lamm_cy(P),F))) ) ).

% finite_image_set
tff(fact_3287_finite__image__set,axiom,
    ! [P: fun_complex_o,F: fun_complex_complex] :
      ( finite3207457112153483333omplex(collect_complex(P))
     => finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_cz(P),F))) ) ).

% finite_image_set
tff(fact_3288_finite__image__set,axiom,
    ! [P: fun_real_o,F: fun_real_list_nat] :
      ( finite_finite_real(collect_real(P))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun5562329742113357962_nat_o(aTP_Lamm_da(P),F))) ) ).

% finite_image_set
tff(fact_3289_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_real_o,F: fun_re7681423839302404990l_real] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(Q))
       => finite_finite_real(collect_real(aa_fun6833002213068288777real_o(aa_fun8577581014811940695real_o(aTP_Lamm_db(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3290_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_real_o,F: fun_re7428432522468557474al_nat] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(Q))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6822315102276328897_nat_o(aa_fun4096426214187263247_nat_o(aTP_Lamm_dc(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3291_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_real_o,F: fun_re7812779276277946752omplex] :
      ( finite_finite_real(collect_real(P))
     => ( finite_finite_real(collect_real(Q))
       => finite3207457112153483333omplex(collect_complex(aa_fun890640625120065285plex_o(aa_fun6537733294925973843plex_o(aTP_Lamm_dd(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3292_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_nat_o,F: fun_re4238438383274954274t_real] :
      ( finite_finite_real(collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(Q))
       => finite_finite_real(collect_real(aa_fun3910182461874428645real_o(aa_fun8527292551407841495real_o(aTP_Lamm_de(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3293_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_nat_o,F: fun_real_fun_nat_nat] :
      ( finite_finite_real(collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(Q))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1675651493036827037_nat_o(aa_fun957765429596782223_nat_o(aTP_Lamm_df(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3294_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_nat_o,F: fun_re2084383175013312036omplex] :
      ( finite_finite_real(collect_real(P))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(Q))
       => finite3207457112153483333omplex(collect_complex(aa_fun2810168320254036705plex_o(aa_fun467192419720471251plex_o(aTP_Lamm_dg(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3295_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_complex_o,F: fun_re2354894731504631552x_real] :
      ( finite_finite_real(collect_real(P))
     => ( finite3207457112153483333omplex(collect_complex(Q))
       => finite_finite_real(collect_real(aa_fun2419829551541144455real_o(aa_fun4137116173344588375real_o(aTP_Lamm_dh(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3296_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_complex_o,F: fun_re175488254389786404ex_nat] :
      ( finite_finite_real(collect_real(P))
     => ( finite3207457112153483333omplex(collect_complex(Q))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6402932185680802879_nat_o(aa_fun3831867060791545871_nat_o(aTP_Lamm_di(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3297_finite__image__set2,axiom,
    ! [P: fun_real_o,Q: fun_complex_o,F: fun_re7208826350674946562omplex] :
      ( finite_finite_real(collect_real(P))
     => ( finite3207457112153483333omplex(collect_complex(Q))
       => finite3207457112153483333omplex(collect_complex(aa_fun434462017428657539plex_o(aa_fun7128336235388974163plex_o(aTP_Lamm_dj(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3298_finite__image__set2,axiom,
    ! [P: fun_nat_o,Q: fun_real_o,F: fun_na6724076748692969250l_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ( finite_finite_real(collect_real(Q))
       => finite_finite_real(collect_real(aa_fun1426617528609952229real_o(aa_fun7503860880236520883real_o(aTP_Lamm_dk(P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_3299_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ? [X4: real] :
            ( member_real(X4,A4)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3300_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A4)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3301_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( finite_finite_rat(A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( member_rat(X4,A4)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3302_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( finite_finite_num(A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X4: num] :
            ( member_num(X4,A4)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3303_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3304_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( finite_finite_int(A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X4: int] :
            ( member_int(X4,A4)
            & ! [Xa: int] :
                ( member_int(Xa,A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X4)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3305_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ? [X4: real] :
            ( member_real(X4,A4)
            & ! [Xa: real] :
                ( member_real(Xa,A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3306_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( finite1152437895449049373et_nat(A4)
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A4)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A4)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3307_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( finite_finite_rat(A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( member_rat(X4,A4)
            & ! [Xa: rat] :
                ( member_rat(Xa,A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3308_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( finite_finite_num(A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X4: num] :
            ( member_num(X4,A4)
            & ! [Xa: num] :
                ( member_num(Xa,A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3309_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3310_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( finite_finite_int(A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X4: int] :
            ( member_int(X4,A4)
            & ! [Xa: int] :
                ( member_int(Xa,A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa)
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3311_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),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) )
                 => ~ 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) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) )
                     => ( ( 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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3312_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3313_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),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)
                   => ~ 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)
                     => ~ 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,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) )
                     => ( ( (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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3314_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3315_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3316_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3317_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList,Summary),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3318_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3319_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q3),zero_z5237406670263579293d_enat) = Q3 ).

% max_enat_simps(2)
tff(fact_3320_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_3321_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_3322_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_3323_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_3324_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_3325_imult__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_3326_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3327_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb2),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( (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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3328_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_3329_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_3330_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ finite_finite_rat(set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_3331_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ finite_finite_real(set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_3332_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_3333_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3334_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3335_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3336_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_3337_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
        | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
          & aa_set_nat_o(ord_less_eq_set_nat(B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3338_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3339_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3340_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3341_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ord_less_eq_set_int(set_or1266510415728281911st_int(A,B),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3342_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3343_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3344_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3345_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3346_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3347_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3348_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_3349_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3350_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3351_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3352_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3353_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3354_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_3355_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_3356_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_3357_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_3358_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_3359_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_eq_set_nat(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3360_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_3361_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_3362_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3363_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( member_int(I,set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3364_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_3365_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
          & ~ aa_set_nat_o(ord_less_eq_set_nat(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3366_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_3367_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_3368_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3369_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_3370_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_3371_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M6: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X4: nat] :
            ( aa_nat_o(P,X4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M6) )
       => ~ ! [M: nat] :
              ( aa_nat_o(P,M)
             => ~ ! [X: nat] :
                    ( aa_nat_o(P,X)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_3372_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc8114805097830739791at_num] :
      ~ ! [F2: fun_nat_fun_num_num,A3: nat,B3: nat,Acc: num] : Xb != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A3,product_Pair_nat_num(B3,Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_3373_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_3374_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_3375_times__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K2),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3376_minus__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),zero_zero_int) = K2 ).

% minus_int_code(1)
tff(fact_3377_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_3378_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_3379_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_3380_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_3381_finite__nat__set__iff__bounded,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3382_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Nb) )
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% bounded_nat_set_is_finite
tff(fact_3383_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_3384_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_dl(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_3385_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_dm(F),U))) ) ).

% finite_less_ub
tff(fact_3386_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ finite_finite_rat(set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_3387_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ finite_finite_real(set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_3388_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K2: 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),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),J)) ) ) ).

% zmult_zless_mono2
tff(fact_3389_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_3390_int__ge__induct,axiom,
    ! [K2: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I)
     => ( aa_int_o(P,K2)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),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_3391_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_3392_int__gr__induct,axiom,
    ! [K2: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),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_3393_int__le__induct,axiom,
    ! [I: int,K2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K2)
     => ( aa_int_o(P,K2)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K2)
             => ( 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_3394_int__less__induct,axiom,
    ! [I: int,K2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K2)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2)
             => ( 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_3395_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C),A)
            & aa_set_nat_o(ord_less_eq_set_nat(B),D)
            & ( aa_set_nat_o(ord_less_set_nat(C),A)
              | aa_set_nat_o(ord_less_set_nat(B),D) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3396_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3397_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3398_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3399_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ord_less_set_int(set_or1266510415728281911st_int(A,B),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3400_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3401_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_3402_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3403_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_3404_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_3405_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_3406_int__induct,axiom,
    ! [P: fun_int_o,K2: int,I: int] :
      ( aa_int_o(P,K2)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),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),K2)
             => ( 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_3407_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_3408_cppi,axiom,
    ! [D3: int,P: fun_int_o,P3: fun_int_o,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
           => ( aa_int_o(P,X4)
            <=> aa_int_o(P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A4)
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X4)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
         => ( ! [X4: int,K: int] :
                ( aa_int_o(P3,X4)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P3,X3) )
                | ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa2: int] :
                        ( member_int(Xa2,A4)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X3)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3409_cpmi,axiom,
    ! [D3: int,P: fun_int_o,P3: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
           => ( aa_int_o(P,X4)
            <=> aa_int_o(P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B4)
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X4)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
         => ( ! [X4: int,K: int] :
                ( aa_int_o(P3,X4)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P3,X3) )
                | ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa2: int] :
                        ( member_int(Xa2,B4)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa2),X3)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3410_aset_I8_J,axiom,
    ! [D3: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) ) ) ).

% aset(8)
tff(fact_3411_aset_I6_J,axiom,
    ! [D3: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int),A4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_3412_bset_I8_J,axiom,
    ! [D3: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int),B4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) ) ) ) ).

% bset(8)
tff(fact_3413_bset_I6_J,axiom,
    ! [D3: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)),Ta) ) ) ) ).

% bset(6)
tff(fact_3414_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dn(Nb),C))) ) ).

% finite_nth_roots
tff(fact_3415_aset_I7_J,axiom,
    ! [D3: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) ) ) ).

% aset(7)
tff(fact_3416_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% minf(7)
tff(fact_3417_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% minf(7)
tff(fact_3418_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% minf(7)
tff(fact_3419_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% minf(7)
tff(fact_3420_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% minf(7)
tff(fact_3421_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% minf(5)
tff(fact_3422_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% minf(5)
tff(fact_3423_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% minf(5)
tff(fact_3424_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% minf(5)
tff(fact_3425_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% minf(5)
tff(fact_3426_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3427_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3428_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3429_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3430_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_3431_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3432_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3433_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3434_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3435_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_3436_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3437_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
           => ( aa_rat_o(Q,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3438_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
           => ( aa_num_o(Q,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3439_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3440_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_3441_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z4)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3442_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z4)
           => ( aa_rat_o(Q,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3443_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z4)
           => ( aa_num_o(Q,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3444_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z4)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3445_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_3446_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% pinf(7)
tff(fact_3447_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% pinf(7)
tff(fact_3448_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% pinf(7)
tff(fact_3449_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% pinf(7)
tff(fact_3450_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% pinf(7)
tff(fact_3451_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% pinf(5)
tff(fact_3452_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% pinf(5)
tff(fact_3453_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% pinf(5)
tff(fact_3454_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% pinf(5)
tff(fact_3455_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% pinf(5)
tff(fact_3456_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3457_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3458_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3459_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3460_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_3461_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3462_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3463_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3464_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3465_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_3466_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3467_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
           => ( aa_rat_o(Q,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3468_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
           => ( aa_num_o(Q,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3469_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3470_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_3471_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X4)
           => ( aa_real_o(Q,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3472_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X4)
           => ( aa_rat_o(Q,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3473_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X4)
           => ( aa_num_o(Q,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3474_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X4)
           => ( aa_nat_o(Q,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3475_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
           => ( aa_int_o(Q,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_3476_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% minf(8)
tff(fact_3477_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% minf(8)
tff(fact_3478_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% minf(8)
tff(fact_3479_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% minf(8)
tff(fact_3480_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% minf(8)
tff(fact_3481_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% minf(6)
tff(fact_3482_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% minf(6)
tff(fact_3483_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% minf(6)
tff(fact_3484_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% minf(6)
tff(fact_3485_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% minf(6)
tff(fact_3486_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% pinf(8)
tff(fact_3487_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% pinf(8)
tff(fact_3488_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% pinf(8)
tff(fact_3489_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% pinf(8)
tff(fact_3490_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% pinf(8)
tff(fact_3491_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% pinf(6)
tff(fact_3492_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% pinf(6)
tff(fact_3493_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% pinf(6)
tff(fact_3494_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% pinf(6)
tff(fact_3495_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% pinf(6)
tff(fact_3496_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D3: real,Q: fun_real_o] :
      ( ! [X4: real,K: real] :
          ( aa_real_o(P,X4)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3))) )
     => ( ! [X4: real,K: real] :
            ( aa_real_o(Q,X4)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              | aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))
              | aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_3497_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X4: rat,K: rat] :
          ( aa_rat_o(P,X4)
        <=> aa_rat_o(P,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
     => ( ! [X4: rat,K: rat] :
            ( aa_rat_o(Q,X4)
          <=> aa_rat_o(Q,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              | aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))
              | aa_rat_o(Q,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_3498_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X4: int,K: int] :
          ( aa_int_o(P,X4)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
     => ( ! [X4: int,K: int] :
            ( aa_int_o(Q,X4)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              | aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_3499_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D3: real,Q: fun_real_o] :
      ( ! [X4: real,K: real] :
          ( aa_real_o(P,X4)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3))) )
     => ( ! [X4: real,K: real] :
            ( aa_real_o(Q,X4)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),D3))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              & aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))
              & aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_3500_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q: fun_rat_o] :
      ( ! [X4: rat,K: rat] :
          ( aa_rat_o(P,X4)
        <=> aa_rat_o(P,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
     => ( ! [X4: rat,K: rat] :
            ( aa_rat_o(Q,X4)
          <=> aa_rat_o(Q,minus_minus_rat(X4,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D3))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              & aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3)))
              & aa_rat_o(Q,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_3501_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D3: int,Q: fun_int_o] :
      ( ! [X4: int,K: int] :
          ( aa_int_o(P,X4)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
     => ( ! [X4: int,K: int] :
            ( aa_int_o(Q,X4)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              & aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_3502_bset_I1_J,axiom,
    ! [D3: int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B4)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B4)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X4)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_3503_bset_I2_J,axiom,
    ! [D3: int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B4)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B4)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X4)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_3504_aset_I1_J,axiom,
    ! [D3: int,A4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A4)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A4)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X4)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_3505_aset_I2_J,axiom,
    ! [D3: int,A4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A4)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A4)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X4)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_3506_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)
     => ( ! [X4: int,K: int] :
            ( aa_int_o(P1,X4)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z4)
             => ( aa_int_o(P,X4)
              <=> aa_int_o(P1,X4) ) )
         => ( ? [X_1: int] : aa_int_o(P1,X_1)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_3507_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int,K: int] :
            ( aa_int_o(P3,X4)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X4)
             => ( aa_int_o(P,X4)
              <=> aa_int_o(P3,X4) ) )
         => ( ? [X_1: int] : aa_int_o(P3,X_1)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_3508_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3509_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3510_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)
     => ( ! [X4: int,K: int] :
            ( aa_int_o(P,X4)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X3: int] :
              ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X3) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3511_bset_I3_J,axiom,
    ! [D3: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int),B4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_3512_bset_I4_J,axiom,
    ! [D3: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(Ta,B4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_3513_bset_I5_J,axiom,
    ! [D3: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)),Ta) ) ) ) ).

% bset(5)
tff(fact_3514_bset_I7_J,axiom,
    ! [D3: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(Ta,B4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)) ) ) ) ) ).

% bset(7)
tff(fact_3515_aset_I3_J,axiom,
    ! [D3: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int),A4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_3516_aset_I4_J,axiom,
    ! [D3: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(Ta,A4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_3517_aset_I5_J,axiom,
    ! [D3: int,Ta: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(Ta,A4)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A4)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_3518_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A3: real,B3: real,C2: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A3),B3)
           => ( aa_real_o(aa_real_fun_real_o(P,B3),C2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),C2)
                 => aa_real_o(aa_real_fun_real_o(P,A3),C2) ) ) ) )
       => ( ! [X4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [A3: real,B3: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X4)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A3)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A3),B3) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_3519_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_3520_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(aa_rat_fun_rat_rat(times_times_rat,Z),Xb)),aa_rat_rat(aa_rat_fun_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_3521_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_3522_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_3523_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(aa_rat_fun_rat_rat(times_times_rat,Xb),Z)),aa_rat_rat(aa_rat_fun_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_3524_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_3525_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_3526_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_3527_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_3528_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Y: fun_int_complex] :
      ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_do(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_do(I6),Y)))
       => finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_dp(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3529_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_dq(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_dq(I6),Y)))
       => finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_dr(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3530_prod_Ofinite__Collect__op,axiom,
    ! [I6: 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_ds(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_ds(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_dt(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3531_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Y: fun_complex_complex] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_du(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_du(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dv(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3532_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dw(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dw(I6),Y)))
       => finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_dx(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3533_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dy(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dy(I6),Y)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dz(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3534_prod_Ofinite__Collect__op,axiom,
    ! [I6: 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(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ea(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_eb(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3535_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ec(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ec(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ed(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3536_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_rat,Y: fun_int_rat] :
      ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ee(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ee(I6),Y)))
       => finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_ef(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3537_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_eg(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_eg(I6),Y)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_eh(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_3538_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_3539_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
    <=> ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q3)),R2) )
        & $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) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3540_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_3541_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(aa_rat_fun_rat_rat(times_times_rat,Xb),Z)),aa_rat_rat(aa_rat_fun_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_3542_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_3543_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_3544_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Y: fun_int_complex] :
      ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_ei(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_ei(I6),Y)))
       => finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ej(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3545_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ek(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_ek(I6),Y)))
       => finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_el(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3546_sum_Ofinite__Collect__op,axiom,
    ! [I6: 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_em(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_em(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_en(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3547_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Y: fun_complex_complex] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_eo(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_eo(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_ep(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3548_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_eq(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_eq(I6),Y)))
       => finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_er(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3549_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_es(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_es(I6),Y)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_et(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3550_sum_Ofinite__Collect__op,axiom,
    ! [I6: 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_eu(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_eu(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_ev(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3551_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ew(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ew(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ex(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3552_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_rat,Y: fun_int_rat] :
      ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ey(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ey(I6),Y)))
       => finite_finite_int(collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_ez(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3553_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fa(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_fa(I6),Y)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_fb(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_3554_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M),zero_zero_nat)
     => ( ! [M: 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,M),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_3555_concat__bit__Suc,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Nb),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(K2),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(Nb,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_3556_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3557_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3558_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_3559_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_3560_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3561_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3562_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3563_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_3564_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_3565_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_3566_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
    <=> ( Mb = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3567_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_3568_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_3569_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_3570_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_3571_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_3572_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_3573_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_3574_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),zero_zero_complex) ).

% dvd_0_right
tff(fact_3575_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_3576_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_3577_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_3578_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_3579_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3580_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3581_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3582_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3583_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3584_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3585_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3586_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3587_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3588_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3589_dvd__1__left,axiom,
    ! [K2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K2) ).

% dvd_1_left
tff(fact_3590_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_3591_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3592_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3593_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(B),A)),aa_int_int(divide_divide_int(C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3594_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( ( K2 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3595_concat__bit__0,axiom,
    ! [K2: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K2),L) = L ).

% concat_bit_0
tff(fact_3596_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_3597_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_3598_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_3599_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_3600_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3601_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3602_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3603_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3604_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3605_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3606_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3607_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3608_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3609_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3610_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3611_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3612_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3613_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3614_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3615_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3616_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_3617_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_3618_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_3619_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3620_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3621_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3622_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3623_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3624_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3625_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3626_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3627_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3628_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3629_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3630_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3631_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3632_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3633_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3634_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3635_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_3636_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_3637_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_3638_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_3639_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_3640_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_3641_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3642_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3643_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(divide_divide_int(one_one_int),aa_int_int(divide_divide_int(one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3644_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_3645_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_3646_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_add
tff(fact_3647_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_3648_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ) ).

% div_diff
tff(fact_3649_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_3650_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_3651_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_3652_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(Nb,K2),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_3653_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Nb,K2),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_3654_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_3655_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K2)) = numeral_numeral_real(aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_3656_dbl__simps_I5_J,axiom,
    ! [K2: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K2)) = numeral_numeral_rat(aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_3657_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_3658_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_3659_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3660_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(divide_divide_int(one_one_int),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% unit_mult_div_div
tff(fact_3661_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3662_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3663_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3664_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_3665_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_3666_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3667_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3668_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_3669_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_3670_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_3671_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3672_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3673_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3674_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3675_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3676_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3677_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_3678_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_3679_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_3680_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3681_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_3682_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3683_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3684_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3685_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_3686_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_3687_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_3688_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3689_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3690_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3691_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3692_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3693_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3694_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_3695_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_3696_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_3697_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_3698_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3699_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3700_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3701_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3702_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3703_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3704_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3705_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3706_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3707_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3708_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3709_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3710_even__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3711_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3712_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3713_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_3714_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3715_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3716_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3717_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3718_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3719_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3720_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3721_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3722_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3723_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3724_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3725_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3726_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_3727_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_3728_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_trans
tff(fact_3729_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_3730_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_3731_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_3732_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Y3) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X4),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_3733_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X4: int,Y3: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X4),Y3) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X4),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y3),B) ) ) ) ).

% dvd_productE
tff(fact_3734_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B6: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C5) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B6),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),C) ) ) ).

% division_decomp
tff(fact_3735_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B6: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B6),C5) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B6),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),C) ) ) ).

% division_decomp
tff(fact_3736_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_3737_dvd__0__left,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_3738_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_3739_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_3740_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_3741_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_3742_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) ) ).

% dvdE
tff(fact_3743_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) ) ).

% dvdE
tff(fact_3744_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K: rat] : A != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K) ) ).

% dvdE
tff(fact_3745_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) ) ).

% dvdE
tff(fact_3746_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K) ) ).

% dvdE
tff(fact_3747_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K2: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_3748_dvdI,axiom,
    ! [A: real,B: real,K2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_3749_dvdI,axiom,
    ! [A: rat,B: rat,K2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K2) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_3750_dvdI,axiom,
    ! [A: nat,B: nat,K2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_3751_dvdI,axiom,
    ! [A: int,B: int,K2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_3752_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvd_def
tff(fact_3753_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_3754_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),K3) ) ).

% dvd_def
tff(fact_3755_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_3756_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_3757_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_3758_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult
tff(fact_3759_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult
tff(fact_3760_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_3761_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_3762_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_3763_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult2
tff(fact_3764_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ).

% dvd_mult2
tff(fact_3765_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_3766_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_3767_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_3768_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_3769_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_3770_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_3771_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_3772_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_3773_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% dvd_triv_left
tff(fact_3774_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% dvd_triv_left
tff(fact_3775_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_3776_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_3777_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3778_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3779_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3780_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3781_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3782_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_3783_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_3784_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_3785_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_3786_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_3787_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_3788_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).

% dvd_triv_right
tff(fact_3789_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)) ).

% dvd_triv_right
tff(fact_3790_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_3791_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_3792_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_3793_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,one_one_complex),A) ).

% one_dvd
tff(fact_3794_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_3795_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_3796_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_3797_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_3798_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_3799_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_3800_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_3801_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_3802_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_3803_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_3804_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)) ) ) ).

% dvd_add
tff(fact_3805_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_3806_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ) ).

% dvd_add
tff(fact_3807_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_3808_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_3809_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3810_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3811_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3812_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3813_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3814_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3815_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3816_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3817_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3818_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3819_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),minus_8373710615458151222nteger(Y,Z)) ) ) ).

% dvd_diff
tff(fact_3820_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_3821_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),minus_minus_rat(Y,Z)) ) ) ).

% dvd_diff
tff(fact_3822_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_3823_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B,C)) ) ).

% dvd_diff_commute
tff(fact_3824_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% dvd_diff_commute
tff(fact_3825_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3826_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),A)
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3827_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3828_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3829_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3830_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3831_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3832_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),A)
       => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3833_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3834_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3835_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3836_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(divide_divide_int(A),C) = aa_int_int(divide_divide_int(B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3837_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_div_div_same
tff(fact_3838_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_3839_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),D)),aa_int_int(divide_divide_int(B),D)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_div_div_same
tff(fact_3840_dvd__power__same,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) ) ).

% dvd_power_same
tff(fact_3841_dvd__power__same,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).

% dvd_power_same
tff(fact_3842_dvd__power__same,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ) ).

% dvd_power_same
tff(fact_3843_dvd__power__same,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) ) ).

% dvd_power_same
tff(fact_3844_dvd__power__same,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) ) ).

% dvd_power_same
tff(fact_3845_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_3846_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_3847_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A),B)),C) = aa_int_int(modulo_modulo_int(A),C) ) ) ).

% mod_mod_cancel
tff(fact_3848_dvd__mod,axiom,
    ! [K2: code_integer,Mb: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K2),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K2),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K2),modulo364778990260209775nteger(Mb,Nb)) ) ) ).

% dvd_mod
tff(fact_3849_dvd__mod,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_3850_dvd__mod,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Mb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(modulo_modulo_int(Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_3851_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3852_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3853_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3854_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3855_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3856_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3857_dvd__diff__nat,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_3858_zdvd__zdiffD,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),Nb))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_3859_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_3860_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X4: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_3861_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X4: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y3)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3862_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X4: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_3863_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_le211207098394363844omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_fc,A)),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_fc,B)))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ).

% subset_divisors_dvd
tff(fact_3864_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_fd,A)),collect_real(aa_real_fun_real_o(aTP_Lamm_fd,B)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% subset_divisors_dvd
tff(fact_3865_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_fe,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_fe,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_3866_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ff,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ff,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_3867_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fg,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fg,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_3868_concat__bit__assoc,axiom,
    ! [Nb: nat,K2: int,Mb: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(Nb,K2),aa_int_int(bit_concat_bit(Mb,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb),aa_int_int(bit_concat_bit(Nb,K2),L)),R2) ).

% concat_bit_assoc
tff(fact_3869_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_less_set_complex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_fc,A)),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_fc,B)))
    <=> ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
        & ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3870_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_fd,A)),collect_real(aa_real_fun_real_o(aTP_Lamm_fd,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3871_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fg,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fg,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3872_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_fe,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_fe,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3873_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ff,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ff,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3874_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_3875_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_3876_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_3877_pinf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% pinf(9)
tff(fact_3878_pinf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% pinf(9)
tff(fact_3879_pinf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2)) ) ) ).

% pinf(9)
tff(fact_3880_pinf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% pinf(9)
tff(fact_3881_pinf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% pinf(9)
tff(fact_3882_pinf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% pinf(10)
tff(fact_3883_pinf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% pinf(10)
tff(fact_3884_pinf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2)) ) ) ).

% pinf(10)
tff(fact_3885_pinf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% pinf(10)
tff(fact_3886_pinf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% pinf(10)
tff(fact_3887_minf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% minf(9)
tff(fact_3888_minf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% minf(9)
tff(fact_3889_minf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2)) ) ) ).

% minf(9)
tff(fact_3890_minf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% minf(9)
tff(fact_3891_minf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% minf(9)
tff(fact_3892_minf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% minf(10)
tff(fact_3893_minf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% minf(10)
tff(fact_3894_minf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),S2)) ) ) ).

% minf(10)
tff(fact_3895_minf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% minf(10)
tff(fact_3896_minf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% minf(10)
tff(fact_3897_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3898_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3899_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3900_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( divide_divide_rat(A,B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3901_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3902_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3903_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_3904_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_3905_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_3906_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3907_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3908_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3909_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3910_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3911_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3912_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3913_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3914_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3915_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3916_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3917_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3918_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3919_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3920_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3921_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3922_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3923_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3924_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_3925_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3926_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(B),C)),A) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3927_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_3928_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3929_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3930_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3931_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3932_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3933_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3934_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3935_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3936_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_3937_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3938_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3939_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3940_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3941_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(C),D)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3942_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3943_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3944_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3945_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3946_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3947_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3948_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3949_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3950_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3951_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3952_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3953_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3954_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3955_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3956_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),C)),aa_int_int(divide_divide_int(B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3957_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_3958_div__power,axiom,
    ! [B: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_3959_div__power,axiom,
    ! [B: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(power_power_int(aa_int_int(divide_divide_int(A),B)),Nb) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% div_power
tff(fact_3960_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3961_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3962_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3963_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3964_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3965_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3966_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3967_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3968_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3969_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3970_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3971_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3972_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3973_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3974_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3975_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3976_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3977_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3978_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3979_dvd__power__le,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat,Mb: 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,Nb),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3980_dvd__power__le,axiom,
    ! [Xb: nat,Y: nat,Nb: nat,Mb: 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,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3981_dvd__power__le,axiom,
    ! [Xb: real,Y: real,Nb: nat,Mb: 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,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3982_dvd__power__le,axiom,
    ! [Xb: int,Y: int,Nb: nat,Mb: 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,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3983_dvd__power__le,axiom,
    ! [Xb: complex,Y: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_3984_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B)) ) ).

% mod_eq_dvd_iff
tff(fact_3985_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_3986_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_3987_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_3988_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(modulo_modulo_int(A),B))) ).

% dvd_minus_mod
tff(fact_3989_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_3990_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X4: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3991_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_3992_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_3993_dvd__diffD1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_3994_dvd__diffD,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Mb) ) ) ) ).

% dvd_diffD
tff(fact_3995_zdvd__mono,axiom,
    ! [K2: int,Mb: int,Ta: int] :
      ( ( K2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Ta)) ) ) ).

% zdvd_mono
tff(fact_3996_zdvd__mult__cancel,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Nb))
     => ( ( K2 != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_3997_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_3998_dbl__def,axiom,
    ! [Xb: rat] : neg_numeral_dbl_rat(Xb) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Xb) ).

% dbl_def
tff(fact_3999_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_4000_zdvd__reduce,axiom,
    ! [K2: int,Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Mb)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Nb) ) ).

% zdvd_reduce
tff(fact_4001_zdvd__period,axiom,
    ! [A: int,D: int,Xb: int,Ta: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Ta))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_4002_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fg,Mb))) ) ).

% finite_divisors_nat
tff(fact_4003_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).

% unit_dvdE
tff(fact_4004_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).

% unit_dvdE
tff(fact_4005_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).

% unit_dvdE
tff(fact_4006_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X3: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X3))
    <=> ? [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X3,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4007_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X3: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X3))
    <=> ? [X3: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X3),zero_zero_complex))
          & aa_complex_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4008_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X3: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X3))
    <=> ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),zero_zero_real))
          & aa_real_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4009_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X3: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X3))
    <=> ? [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),zero_zero_rat))
          & aa_rat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4010_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X3: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3))
    <=> ? [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat))
          & aa_nat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4011_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X3: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3))
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int))
          & aa_int_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4012_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4013_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4014_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(divide_divide_int(B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4015_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4016_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4017_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4018_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4019_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4020_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4021_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4022_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4023_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(divide_divide_int(B),A) = aa_int_int(divide_divide_int(D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4024_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4025_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4026_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4027_even__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_4028_even__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_4029_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_4030_inf__period_I4_J,axiom,
    ! [D: code_integer,D3: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D3)
     => ! [X: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),Ta)) ) ) ).

% inf_period(4)
tff(fact_4031_inf__period_I4_J,axiom,
    ! [D: real,D3: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4032_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4033_inf__period_I4_J,axiom,
    ! [D: int,D3: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4034_inf__period_I3_J,axiom,
    ! [D: code_integer,D3: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D3)
     => ! [X: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),Ta)) ) ) ).

% inf_period(3)
tff(fact_4035_inf__period_I3_J,axiom,
    ! [D: real,D3: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4036_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(X,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D3))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4037_inf__period_I3_J,axiom,
    ! [D: int,D3: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4038_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4039_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4040_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(divide_divide_int(A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4041_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4042_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4043_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(divide_divide_int(C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4044_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_4045_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4046_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4047_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_4048_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4049_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A),B)),C) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4050_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_4051_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4052_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(divide_divide_int(B),C)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4053_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4054_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4055_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4056_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_4057_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_4058_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_4059_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4060_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4061_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4062_dvd__imp__le,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% dvd_imp_le
tff(fact_4063_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_4064_dvd__mult__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_4065_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_4066_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4067_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_4068_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_4069_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_4070_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B3),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4071_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4072_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B3 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4073_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4074_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4075_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4076_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4077_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4078_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4079_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_4080_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_4081_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_4082_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_4083_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_4084_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_4085_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_4086_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_4087_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_4088_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) )
        & ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4089_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4090_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
        & ( aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4091_dvd__power__iff,axiom,
    ! [Xb: code_integer,Mb: nat,Nb: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_4092_dvd__power__iff,axiom,
    ! [Xb: nat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_4093_dvd__power__iff,axiom,
    ! [Xb: int,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_4094_dvd__power,axiom,
    ! [Nb: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)) ) ).

% dvd_power
tff(fact_4095_dvd__power,axiom,
    ! [Nb: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ).

% dvd_power
tff(fact_4096_dvd__power,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).

% dvd_power
tff(fact_4097_dvd__power,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ).

% dvd_power
tff(fact_4098_dvd__power,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),Nb)) ) ).

% dvd_power
tff(fact_4099_dvd__power,axiom,
    ! [Nb: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_nat_complex(power_power_complex(Xb),Nb)) ) ).

% dvd_power
tff(fact_4100_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_4101_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_4102_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_4103_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_4104_dvd__minus__add,axiom,
    ! [Q3: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_4105_power__dvd__imp__le,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
     => ( 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,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_4106_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4107_aset_I10_J,axiom,
    ! [D: int,D3: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)),Ta)) ) ) ) ).

% aset(10)
tff(fact_4108_aset_I9_J,axiom,
    ! [D: int,D3: int,A4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A4)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D3)),Ta)) ) ) ) ).

% aset(9)
tff(fact_4109_bset_I10_J,axiom,
    ! [D: int,D3: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)),Ta)) ) ) ) ).

% bset(10)
tff(fact_4110_bset_I9_J,axiom,
    ! [D: int,D3: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D3)),Ta)) ) ) ) ).

% bset(9)
tff(fact_4111_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4112_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4113_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4114_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4115_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4116_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_4117_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4118_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4119_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_4120_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_4121_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_4122_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_4123_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_4124_dvd__power__iff__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_4125_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4126_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4127_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_4128_even__set__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4129_even__set__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4130_even__set__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_4131_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4132_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Mb),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4133_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_4134_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_diff_iff
tff(fact_4135_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_4136_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_4137_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_4138_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_4139_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_4140_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_4141_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_4142_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_4143_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_4144_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4145_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4146_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_4147_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_4148_zero__le__odd__power,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_4149_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_4150_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_even_power
tff(fact_4151_zero__le__even__power,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_4152_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_even_power
tff(fact_4153_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4154_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4155_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_4156_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B3),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),zero_zero_nat)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_4157_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4158_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4159_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_4160_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4161_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4162_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_4163_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_4164_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_4165_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_4166_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_4167_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4168_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4169_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4170_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4171_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4172_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4173_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( finite_finite_real(S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X4: real] :
            ( member_real(X4,S3)
            & ~ ? [Xa: real] :
                  ( member_real(Xa,S3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4174_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( finite_finite_rat(S3)
     => ( ( S3 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( member_rat(X4,S3)
            & ~ ? [Xa: rat] :
                  ( member_rat(Xa,S3)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4175_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( finite_finite_num(S3)
     => ( ( S3 != bot_bot_set_num )
       => ? [X4: num] :
            ( member_num(X4,S3)
            & ~ ? [Xa: num] :
                  ( member_num(Xa,S3)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4176_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),S3)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S3)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4177_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
     => ( ( S3 != bot_bot_set_int )
       => ? [X4: int] :
            ( member_int(X4,S3)
            & ~ ? [Xa: int] :
                  ( member_int(Xa,S3)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X4) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4178_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( member_real(X4,X6)
           => ? [Xa: real] :
                ( member_real(Xa,X6)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Xa) ) )
       => ~ finite_finite_real(X6) ) ) ).

% infinite_growing
tff(fact_4179_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( member_rat(X4,X6)
           => ? [Xa: rat] :
                ( member_rat(Xa,X6)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Xa) ) )
       => ~ finite_finite_rat(X6) ) ) ).

% infinite_growing
tff(fact_4180_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( member_num(X4,X6)
           => ? [Xa: num] :
                ( member_num(Xa,X6)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Xa) ) )
       => ~ finite_finite_num(X6) ) ) ).

% infinite_growing
tff(fact_4181_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),X6)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X6)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X6) ) ) ).

% infinite_growing
tff(fact_4182_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( member_int(X4,X6)
           => ? [Xa: int] :
                ( member_int(Xa,X6)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Xa) ) )
       => ~ finite_finite_int(X6) ) ) ).

% infinite_growing
tff(fact_4183_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) ) )
         => ~ ! [Va3: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
               => ( 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,Va3))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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_4184_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4185_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4186_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4187_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_4188_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,Xb))
    <=> aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_4189_diff__shunt__var,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xb,Y) = bot_bot_set_int )
    <=> ord_less_eq_set_int(Xb,Y) ) ).

% diff_shunt_var
tff(fact_4190_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_4191_diff__shunt__var,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xb,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).

% diff_shunt_var
tff(fact_4192_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4193_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4194_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4195_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4196_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4197_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)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_4198_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_ao,set_or1266510415728281911st_int(Mb,Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_4199_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_o(P,Xb)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I)) ) ) ).

% intind
tff(fact_4200_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_int_o(P,Xb)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xb)),I)) ) ) ).

% intind
tff(fact_4201_intind,axiom,
    ! [I: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_VEBT_VEBT_o(P,Xb)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I)) ) ) ).

% intind
tff(fact_4202_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_4203_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_4204_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_4205_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4206_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex($false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_4207_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4208_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4209_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat($false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4210_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4211_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger($false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_4212_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4213_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4214_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4215_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4216_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_4217_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = zero_z3403309356797280102nteger )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4218_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_4219_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_4220_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_4221_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_4222_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4223_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4224_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4225_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4226_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4227_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_4228_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4229_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_4230_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_4231_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4232_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat($true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4233_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_4234_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_4235_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_4236_replicate__eq__replicate,axiom,
    ! [Mb: nat,Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Mb,Xb) = replicate_VEBT_VEBT(Nb,Y) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero_nat )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_4237_length__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_4238_length__replicate,axiom,
    ! [Nb: nat,Xb: $o] : size_size_list_o(replicate_o(Nb,(Xb))) = Nb ).

% length_replicate
tff(fact_4239_length__replicate,axiom,
    ! [Nb: nat,Xb: nat] : size_size_list_nat(replicate_nat(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_4240_length__replicate,axiom,
    ! [Nb: nat,Xb: int] : size_size_list_int(replicate_int(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_4241_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_4242_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_4243_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_4244_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_4245_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_4246_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_4247_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_4248_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4249_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_4250_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_4251_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_4252_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_4253_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_4254_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_4255_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_4256_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4257_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4258_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_4259_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_4260_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = minus_minus_rat(one_one_rat,zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_4261_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_4262_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n356916108424825756nteger(~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((P))) ).

% of_bool_not_iff
tff(fact_4263_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = zero_n2687167440665602831ol_nat(Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_4264_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_4265_signed__take__bit__numeral__of__1,axiom,
    ! [K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K2)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_4266_in__set__replicate,axiom,
    ! [Xb: real,Nb: nat,Y: real] :
      ( member_real(Xb,set_real2(replicate_real(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4267_in__set__replicate,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] :
      ( member_complex(Xb,set_complex2(replicate_complex(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4268_in__set__replicate,axiom,
    ! [Xb: product_prod_nat_nat,Nb: nat,Y: product_prod_nat_nat] :
      ( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4269_in__set__replicate,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( member_int(Xb,set_int2(replicate_int(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4270_in__set__replicate,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4271_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4272_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X3: int] :
          ( member_int(X3,set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4273_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X3) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4274_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4275_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4276_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X3) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4277_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4278_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_4279_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_4280_nth__replicate,axiom,
    ! [I: nat,Nb: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I) = Xb ) ) ).

% nth_replicate
tff(fact_4281_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_4282_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_4283_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_4284_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_4285_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_n2687167440665602831ol_nat((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_4286_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_4287_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_4288_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat((B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_4289_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(divide_divide_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_4290_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4291_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).

% set_decode_0
tff(fact_4292_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4293_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4294_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4295_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4296_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4297_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4298_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4299_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4300_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4301_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_4302_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( zero_n2687167440665602831ol_nat((P2)) = zero_n2687167440665602831ol_nat((Q3)) )
    <=> ( (P2)
      <=> (Q3) ) ) ).

% of_bool_eq_iff
tff(fact_4303_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P2)) = aa_o_int(zero_n2684676970156552555ol_int,(Q3)) )
    <=> ( (P2)
      <=> (Q3) ) ) ).

% of_bool_eq_iff
tff(fact_4304_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( zero_n356916108424825756nteger((P2)) = zero_n356916108424825756nteger((Q3)) )
    <=> ( (P2)
      <=> (Q3) ) ) ).

% of_bool_eq_iff
tff(fact_4305_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_4306_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2052037380579107095ol_rat(( (P)
        & (Q) )) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q))) ).

% of_bool_conj
tff(fact_4307_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_4308_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_4309_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q))) ).

% of_bool_conj
tff(fact_4310_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_4311_signed__take__bit__mult,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K2),L)) ).

% signed_take_bit_mult
tff(fact_4312_signed__take__bit__add,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ).

% signed_take_bit_add
tff(fact_4313_signed__take__bit__diff,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)) ).

% signed_take_bit_diff
tff(fact_4314_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_4315_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5)) ) ).

% sum_mono
tff(fact_4316_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_4317_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5)) ) ).

% sum_mono
tff(fact_4318_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5)) ) ).

% sum_mono
tff(fact_4319_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( member_int(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_4320_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_4321_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5)) ) ).

% sum_mono
tff(fact_4322_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5)) ) ).

% sum_mono
tff(fact_4323_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_4324_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I2: complex] :
          ( member_complex(I2,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5)) ) ).

% sum_mono
tff(fact_4325_sum__product,axiom,
    ! [F: fun_int_int,A4: set_int,G: fun_int_int,B4: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(G,B4)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_fi(F),G),B4),A4) ).

% sum_product
tff(fact_4326_sum__product,axiom,
    ! [F: fun_complex_complex,A4: set_complex,G: fun_complex_complex,B4: set_complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A4)),groups7754918857620584856omplex(G,B4)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_fk(F),G),B4),A4) ).

% sum_product
tff(fact_4327_sum__product,axiom,
    ! [F: fun_nat_nat,A4: set_nat,G: fun_nat_nat,B4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_fm(F),G),B4)),A4) ).

% sum_product
tff(fact_4328_sum__product,axiom,
    ! [F: fun_nat_real,A4: set_nat,G: fun_nat_real,B4: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A4)),groups6591440286371151544t_real(G,B4)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_fo(F),G),B4),A4) ).

% sum_product
tff(fact_4329_sum__distrib__right,axiom,
    ! [F: fun_int_int,A4: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A4)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_fp(F),R2),A4) ).

% sum_distrib_right
tff(fact_4330_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A4: set_complex,R2: complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A4)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_fq(F),R2),A4) ).

% sum_distrib_right
tff(fact_4331_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A4: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fr(F),R2)),A4) ).

% sum_distrib_right
tff(fact_4332_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A4: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A4)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fs(F),R2),A4) ).

% sum_distrib_right
tff(fact_4333_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A4: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A4)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ft(R2),F),A4) ).

% sum_distrib_left
tff(fact_4334_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A4: set_complex] : aa_complex_complex(times_times_complex(R2),groups7754918857620584856omplex(F,A4)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fu(R2),F),A4) ).

% sum_distrib_left
tff(fact_4335_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fv(R2),F)),A4) ).

% sum_distrib_left
tff(fact_4336_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A4: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),groups6591440286371151544t_real(F,A4)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fw(R2),F),A4) ).

% sum_distrib_left
tff(fact_4337_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fx(G),H),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A4)),groups4538972089207619220nt_int(H,A4)) ).

% sum.distrib
tff(fact_4338_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A4: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fy(G),H),A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A4)),groups7754918857620584856omplex(H,A4)) ).

% sum.distrib
tff(fact_4339_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fz(G),H)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A4)) ).

% sum.distrib
tff(fact_4340_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A4: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(G),H),A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A4)),groups6591440286371151544t_real(H,A4)) ).

% sum.distrib
tff(fact_4341_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A4: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gb(F),G),A4) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(G,A4)) ).

% sum_subtractf
tff(fact_4342_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A4: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_gc(F),G),A4) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A4)),groups7754918857620584856omplex(G,A4)) ).

% sum_subtractf
tff(fact_4343_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A4: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gd(F),G),A4) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A4)),groups6591440286371151544t_real(G,A4)) ).

% sum_subtractf
tff(fact_4344_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A4: set_complex,R2: complex] : aa_complex_complex(divide1717551699836669952omplex(groups7754918857620584856omplex(F,A4)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ge(F),R2),A4) ).

% sum_divide_distrib
tff(fact_4345_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A4: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups6591440286371151544t_real(F,A4)),R2) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_gf(F),R2),A4) ).

% sum_divide_distrib
tff(fact_4346_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A4: set_int] : aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_gg(F),A),A4)),A) = aa_int_int(modulo_modulo_int(groups4538972089207619220nt_int(F,A4)),A) ).

% mod_sum_eq
tff(fact_4347_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A4: 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_gh(F),A)),A4)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),A) ).

% mod_sum_eq
tff(fact_4348_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A4)) ) ).

% sum_nonneg
tff(fact_4349_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A4)) ) ).

% sum_nonneg
tff(fact_4350_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A4)) ) ).

% sum_nonneg
tff(fact_4351_sum__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4352_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4353_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4354_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4355_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_4356_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_4357_sum__nonneg,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_4358_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_4359_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_4360_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X4)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_4361_sum__nonpos,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4362_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4363_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X4)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4364_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4365_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_4366_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_4367_sum__nonpos,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X4)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_4368_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I6: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I6) = groups3906332499630173760nt_rat(G,I6) )
     => ( ! [I2: int] :
            ( member_int(I2,I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
       => ( member_int(I,I6)
         => ( finite_finite_int(I6)
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4369_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I6: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I6) = groups1300246762558778688al_rat(G,I6) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => 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,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4370_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I6: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I6) = groups2906978787729119204at_rat(G,I6) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => 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),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4371_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I6: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I6) = groups5058264527183730370ex_rat(G,I6) )
     => ( ! [I2: complex] :
            ( member_complex(I2,I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
       => ( member_complex(I,I6)
         => ( finite3207457112153483333omplex(I6)
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4372_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I6) = groups4541462559716669496nt_nat(G,I6) )
     => ( ! [I2: int] :
            ( member_int(I2,I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
       => ( member_int(I,I6)
         => ( finite_finite_int(I6)
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4373_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I6) = groups1935376822645274424al_nat(G,I6) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => 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,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4374_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I6) = groups5693394587270226106ex_nat(G,I6) )
     => ( ! [I2: complex] :
            ( member_complex(I2,I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
       => ( member_complex(I,I6)
         => ( finite3207457112153483333omplex(I6)
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4375_sum__mono__inv,axiom,
    ! [F: fun_real_int,I6: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I6) = groups1932886352136224148al_int(G,I6) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => 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,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4376_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I6: set_nat,G: fun_nat_int,I: nat] :
      ( ( groups3539618377306564664at_int(F,I6) = groups3539618377306564664at_int(G,I6) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => 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),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4377_sum__mono__inv,axiom,
    ! [F: fun_complex_int,I6: set_complex,G: fun_complex_int,I: complex] :
      ( ( groups5690904116761175830ex_int(F,I6) = groups5690904116761175830ex_int(G,I6) )
     => ( ! [I2: complex] :
            ( member_complex(I2,I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
       => ( member_complex(I,I6)
         => ( finite3207457112153483333omplex(I6)
           => ( aa_complex_int(F,I) = aa_complex_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4378_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_4379_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_4380_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_4381_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_4382_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P))) ).

% zero_less_eq_of_bool
tff(fact_4383_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_4384_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_4385_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_4386_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_4387_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_4388_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_4389_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_4390_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_4391_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2687167440665602831ol_nat((P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_4392_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_4393_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_4394_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_4395_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_4396_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_4397_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_4398_split__of__bool,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ( ( (P2)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_4399_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ( ( (P2)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P2)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_4400_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_4401_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_4402_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_4403_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_4404_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P2)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_4405_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P2)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_4406_finite__set__decode,axiom,
    ! [Nb: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(nat_set_decode,Nb)) ).

% finite_set_decode
tff(fact_4407_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y3: real] :
            ( member_real(Y3,set_real2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_real(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4408_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y3: complex] :
            ( member_complex(Y3,set_complex2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_complex(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4409_replicate__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Nb: nat,Xb: product_prod_nat_nat] :
      ( ( size_s5460976970255530739at_nat(Xs) = Nb )
     => ( ! [Y3: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(Y3,set_Pr5648618587558075414at_nat(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replic4235873036481779905at_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4410_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y3: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y3,set_VEBT_VEBT2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4411_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y3: $o] :
            ( member_o((Y3),set_o2(Xs))
           => ( (Y3)
            <=> (Xb) ) )
       => ( Xs = replicate_o(Nb,(Xb)) ) ) ) ).

% replicate_eqI
tff(fact_4412_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),set_nat2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4413_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y3: int] :
            ( member_int(Y3,set_int2(Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate_int(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4414_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => ( X4 = Xb ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_4415_replicate__length__same,axiom,
    ! [Xs: list_o,Xb: $o] :
      ( ! [X4: $o] :
          ( member_o((X4),set_o2(Xs))
         => ( (X4)
          <=> (Xb) ) )
     => ( replicate_o(size_size_list_o(Xs),(Xb)) = Xs ) ) ).

% replicate_length_same
tff(fact_4416_replicate__length__same,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => ( X4 = Xb ) )
     => ( replicate_nat(size_size_list_nat(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_4417_replicate__length__same,axiom,
    ! [Xs: list_int,Xb: int] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => ( X4 = Xb ) )
     => ( replicate_int(size_size_list_int(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_4418_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X4: complex] :
              ( member_complex(X4,Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4)) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_complex(I,Xa) = X4 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4419_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S2)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I,Xa) = X4 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4420_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X4: complex] :
              ( member_complex(X4,Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_nat(I,Xa) = X4 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4421_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_nat,G: fun_nat_rat,I: fun_nat_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4)) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S2)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_complex(I,Xa) = X4 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S2)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4422_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_complex,G: fun_complex_rat,I: fun_complex_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X4: complex] :
              ( member_complex(X4,Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X4)) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_complex(I,Xa) = X4 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,S2)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4423_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_complex,G: fun_complex_nat,I: fun_complex_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X4: complex] :
              ( member_complex(X4,Ta)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(G,X4)) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_complex(I,Xa) = X4 )
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,Xa)) ) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,S2)),groups5693394587270226106ex_nat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4424_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_nat,G: fun_nat_int,I: fun_nat_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S2)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I,Xa) = X4 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,S2)),groups3539618377306564664at_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4425_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_complex,G: fun_complex_int,I: fun_complex_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X4: complex] :
              ( member_complex(X4,Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(G,X4)) )
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_nat(I,Xa) = X4 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_complex_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,S2)),groups5690904116761175830ex_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4426_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_nat,G: fun_nat_int,I: fun_nat_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(G,X4)) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S2)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_complex(I,Xa) = X4 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X4)),aa_nat_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,S2)),groups3539618377306564664at_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4427_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_complex,G: fun_complex_int,I: fun_complex_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X4: complex] :
              ( member_complex(X4,Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(G,X4)) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_complex(I,Xa) = X4 )
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,Xa)) ) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,S2)),groups5690904116761175830ex_int(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_4428_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( finite_finite_int(A4)
     => ( ! [X4: int] :
            ( member_int(X4,A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)) )
       => ( ( groups8778361861064173332t_real(F,A4) = zero_zero_real )
        <=> ! [X3: int] :
              ( member_int(X3,A4)
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4429_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( finite_finite_real(A4)
     => ( ! [X4: real] :
            ( member_real(X4,A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
       => ( ( groups8097168146408367636l_real(F,A4) = zero_zero_real )
        <=> ! [X3: real] :
              ( member_real(X3,A4)
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4430_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [X4: complex] :
            ( member_complex(X4,A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4)) )
       => ( ( groups5808333547571424918x_real(F,A4) = zero_zero_real )
        <=> ! [X3: complex] :
              ( member_complex(X3,A4)
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4431_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( finite_finite_int(A4)
     => ( ! [X4: int] :
            ( member_int(X4,A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
       => ( ( groups3906332499630173760nt_rat(F,A4) = zero_zero_rat )
        <=> ! [X3: int] :
              ( member_int(X3,A4)
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4432_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( ! [X4: real] :
            ( member_real(X4,A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)) )
       => ( ( groups1300246762558778688al_rat(F,A4) = zero_zero_rat )
        <=> ! [X3: real] :
              ( member_real(X3,A4)
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4433_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)) )
       => ( ( groups2906978787729119204at_rat(F,A4) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4434_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [X4: complex] :
            ( member_complex(X4,A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)) )
       => ( ( groups5058264527183730370ex_rat(F,A4) = zero_zero_rat )
        <=> ! [X3: complex] :
              ( member_complex(X3,A4)
             => ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4435_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( finite_finite_int(A4)
     => ( ! [X4: int] :
            ( member_int(X4,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
       => ( ( groups4541462559716669496nt_nat(F,A4) = zero_zero_nat )
        <=> ! [X3: int] :
              ( member_int(X3,A4)
             => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4436_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( ! [X4: real] :
            ( member_real(X4,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
       => ( ( groups1935376822645274424al_nat(F,A4) = zero_zero_nat )
        <=> ! [X3: real] :
              ( member_real(X3,A4)
             => ( aa_real_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4437_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [X4: complex] :
            ( member_complex(X4,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
       => ( ( groups5693394587270226106ex_nat(F,A4) = zero_zero_nat )
        <=> ! [X3: complex] :
              ( member_complex(X3,A4)
             => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4438_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [X4: complex] :
            ( member_complex(X4,A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4)) )
       => ( ? [X: complex] :
              ( member_complex(X,A4)
              & 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,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4439_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & 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,A4)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4440_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [X4: complex] :
            ( member_complex(X4,A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4)) )
       => ( ? [X: complex] :
              ( member_complex(X,A4)
              & 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,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4441_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [X4: complex] :
            ( member_complex(X4,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4)) )
       => ( ? [X: complex] :
              ( member_complex(X,A4)
              & 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,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4442_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),aa_nat_int(G,X4)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4443_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [X4: complex] :
            ( member_complex(X4,A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X4)),aa_complex_int(G,X4)) )
       => ( ? [X: complex] :
              ( member_complex(X,A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X)),aa_complex_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4444_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_int,G: fun_int_int] :
      ( finite_finite_int(A4)
     => ( ! [X4: int] :
            ( member_int(X4,A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),aa_int_int(G,X4)) )
       => ( ? [X: int] :
              ( member_int(X,A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X)),aa_int_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4445_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),aa_nat_nat(G,X4)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4446_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),aa_nat_real(G,X4)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,A4)),groups6591440286371151544t_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4447_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups2073611262835488442omplex(H,S3)),groups2073611262835488442omplex(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4448_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: 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)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X4)),aa_complex_real(G,X4)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4449_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: 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)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4450_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: 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)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4451_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: 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)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4452_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: 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)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X4)),aa_nat_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S3)),groups3539618377306564664at_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4453_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: 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)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_complex_int(H,X4)),aa_complex_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups5690904116761175830ex_int(H,S3)),groups5690904116761175830ex_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4454_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: 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)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23)) )
       => ( finite_finite_int(S3)
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_int_int(H,X4)),aa_int_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups4538972089207619220nt_int(H,S3)),groups4538972089207619220nt_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4455_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(H,X4)),aa_complex_complex(G,X4)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups7754918857620584856omplex(H,S3)),groups7754918857620584856omplex(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_4456_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_nat,H: fun_nat_nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(H,X4)),aa_nat_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_set_nat_nat(groups3542108847815614940at_nat(H),S3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),S3)) ) ) ) ) ).

% sum.related
tff(fact_4457_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( member_complex(X4,A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4458_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( finite_finite_int(A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( member_int(X4,A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),aa_int_real(G,X4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4459_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( member_real(X4,A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(G,X4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4460_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( member_complex(X4,A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4461_sum__strict__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4462_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( member_int(X4,A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4463_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( member_real(X4,A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(G,X4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4464_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( member_complex(X4,A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4465_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( member_int(X4,A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4466_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( member_real(X4,A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(G,X4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4467_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_4468_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_4469_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_4470_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_4471_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(S4)
     => ( finite_finite_int(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_int(aa_int_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_int(aa_int_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4472_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( finite_finite_int(S4)
     => ( finite_finite_real(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T,T2)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T2))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T2))
                   => member_int(aa_real_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T2)
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups5754745047067104278omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4473_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_int,S3: 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)
     => ( finite_finite_int(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_int(aa_real_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_real(aa_int_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups3049146728041665814omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4474_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_real,S3: 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(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T,T2)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T2))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T2))
                   => member_real(aa_real_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T2)
                       => ( aa_real_complex(H,B3) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4475_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(S4)
     => ( finite_finite_int(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_int(aa_int_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_int(aa_int_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_int_real(H,aa_int_int(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4476_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_real,H: fun_real_real] :
      ( finite_finite_int(S4)
     => ( finite_finite_real(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T,T2)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T2))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T2))
                   => member_int(aa_real_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T2)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_real_real(H,aa_int_real(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4477_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_int,S3: 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)
     => ( finite_finite_int(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_int(aa_real_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_real(aa_int_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_int_real(H,aa_real_int(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4478_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_real,S3: 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(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T,T2)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T2))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T2))
                   => member_real(aa_real_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T2)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4479_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_complex,S3: set_int,I: fun_complex_int,J: fun_int_complex,T: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( finite_finite_int(S4)
     => ( finite3207457112153483333omplex(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_complex(aa_int_complex(J,A3),minus_811609699411566653omplex(T,T2)) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                   => member_int(aa_complex_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( member_complex(B3,T2)
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_complex_real(H,aa_int_complex(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4480_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_complex,S3: 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)
     => ( finite3207457112153483333omplex(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_complex(aa_real_complex(J,A3),minus_811609699411566653omplex(T,T2)) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                   => member_real(aa_complex_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( member_complex(B3,T2)
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_complex_real(H,aa_real_complex(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4481_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_real,I: int] :
      ( finite_finite_int(S2)
     => ( ! [I2: int] :
            ( member_int(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S2) = zero_zero_real )
         => ( member_int(I,S2)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4482_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_real,I: real] :
      ( finite_finite_real(S2)
     => ( ! [I2: real] :
            ( member_real(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S2) = zero_zero_real )
         => ( member_real(I,S2)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4483_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_real,I: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I2: complex] :
            ( member_complex(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S2) = zero_zero_real )
         => ( member_complex(I,S2)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4484_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_rat,I: int] :
      ( finite_finite_int(S2)
     => ( ! [I2: int] :
            ( member_int(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = zero_zero_rat )
         => ( member_int(I,S2)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4485_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_rat,I: real] :
      ( finite_finite_real(S2)
     => ( ! [I2: real] :
            ( member_real(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S2) = zero_zero_rat )
         => ( member_real(I,S2)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4486_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S2) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),S2)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4487_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_rat,I: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I2: complex] :
            ( member_complex(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S2) = zero_zero_rat )
         => ( member_complex(I,S2)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4488_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_nat,I: int] :
      ( finite_finite_int(S2)
     => ( ! [I2: int] :
            ( member_int(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = zero_zero_nat )
         => ( member_int(I,S2)
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4489_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_nat,I: real] :
      ( finite_finite_real(S2)
     => ( ! [I2: real] :
            ( member_real(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S2) = zero_zero_nat )
         => ( member_real(I,S2)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4490_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_nat,I: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I2: complex] :
            ( member_complex(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S2) = zero_zero_nat )
         => ( member_complex(I,S2)
           => ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4491_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_real,B4: real,I: int] :
      ( finite_finite_int(S2)
     => ( ! [I2: int] :
            ( member_int(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S2) = B4 )
         => ( member_int(I,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4492_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_real,B4: real,I: real] :
      ( finite_finite_real(S2)
     => ( ! [I2: real] :
            ( member_real(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S2) = B4 )
         => ( member_real(I,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4493_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_real,B4: real,I: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I2: complex] :
            ( member_complex(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S2) = B4 )
         => ( member_complex(I,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4494_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_rat,B4: rat,I: int] :
      ( finite_finite_int(S2)
     => ( ! [I2: int] :
            ( member_int(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = B4 )
         => ( member_int(I,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4495_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_rat,B4: rat,I: real] :
      ( finite_finite_real(S2)
     => ( ! [I2: real] :
            ( member_real(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S2) = B4 )
         => ( member_real(I,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4496_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_rat,B4: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S2) = B4 )
         => ( aa_set_nat_o(member_nat(I),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4497_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_rat,B4: rat,I: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I2: complex] :
            ( member_complex(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S2) = B4 )
         => ( member_complex(I,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4498_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_nat,B4: nat,I: int] :
      ( finite_finite_int(S2)
     => ( ! [I2: int] :
            ( member_int(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = B4 )
         => ( member_int(I,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4499_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_nat,B4: nat,I: real] :
      ( finite_finite_real(S2)
     => ( ! [I2: real] :
            ( member_real(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S2) = B4 )
         => ( member_real(I,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4500_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_nat,B4: nat,I: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I2: complex] :
            ( member_complex(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S2) = B4 )
         => ( member_complex(I,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4501_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_complex] :
      ( finite_finite_real(A4)
     => ( groups5754745047067104278omplex(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_gi(G)))) = groups5754745047067104278omplex(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4502_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_gj(G)))) = groups8097168146408367636l_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4503_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A4)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_gk(G)))) = groups5808333547571424918x_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4504_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_gl(G)))) = groups1300246762558778688al_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4505_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A4)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_gm(G)))) = groups5058264527183730370ex_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4506_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_gn(G)))) = groups1935376822645274424al_nat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4507_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_go(G)))) = groups5693394587270226106ex_nat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4508_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_int] :
      ( finite_finite_real(A4)
     => ( groups1932886352136224148al_int(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_gp(G)))) = groups1932886352136224148al_int(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4509_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(A4)
     => ( groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_gq(G)))) = groups5690904116761175830ex_int(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4510_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2073611262835488442omplex(G,minus_minus_set_nat(A4,collect_nat(aTP_Lamm_gr(G)))) = groups2073611262835488442omplex(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_4511_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( finite_finite_int(I6)
     => ( member_int(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4512_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4513_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( member_complex(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4514_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( finite_finite_int(I6)
     => ( member_int(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4515_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4516_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( 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),I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4517_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( member_complex(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4518_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: fun_int_nat] :
      ( finite_finite_int(I6)
     => ( member_int(I,I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4519_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: fun_real_nat] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4520_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I,I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I))
         => ( ! [I2: complex] :
                ( member_complex(I2,I6)
               => 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,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_4521_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4522_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4523_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4524_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4525_sum__pos,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4526_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4527_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4528_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4529_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_nat] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4530_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_nat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => 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,I6)) ) ) ) ).

% sum_pos
tff(fact_4531_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_4532_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,A4) = groups3049146728041665814omplex(H,B4) )
              <=> ( groups3049146728041665814omplex(G,C4) = groups3049146728041665814omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4533_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A4) = groups5754745047067104278omplex(H,B4) )
              <=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4534_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A4) = groups8778361861064173332t_real(H,B4) )
              <=> ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4535_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4536_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4537_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A4) = groups3906332499630173760nt_rat(H,B4) )
              <=> ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4538_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4539_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4540_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,A4) = groups4541462559716669496nt_nat(H,B4) )
              <=> ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4541_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4542_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups3049146728041665814omplex(G,C4) = groups3049146728041665814omplex(H,C4) )
               => ( groups3049146728041665814omplex(G,A4) = groups3049146728041665814omplex(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4543_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
               => ( groups5754745047067104278omplex(G,A4) = groups5754745047067104278omplex(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4544_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) )
               => ( groups8778361861064173332t_real(G,A4) = groups8778361861064173332t_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4545_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4546_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4547_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) )
               => ( groups3906332499630173760nt_rat(G,A4) = groups3906332499630173760nt_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4548_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4549_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A4) = groups5058264527183730370ex_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4550_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) )
               => ( groups4541462559716669496nt_nat(G,A4) = groups4541462559716669496nt_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4551_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A4) = groups1935376822645274424al_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4552_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4553_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4554_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4555_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4556_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,S3) = groups2073611262835488442omplex(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4557_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S3) = groups2906978787729119204at_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4558_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S3) = groups3539618377306564664at_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4559_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_int] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_int(G,X4) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,S3) = groups4538972089207619220nt_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4560_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G,S3) = groups7754918857620584856omplex(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4561_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),S3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4562_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4563_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T) = groups5058264527183730370ex_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4564_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4565_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4566_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_complex(G,X4) = zero_zero_complex ) )
         => ( groups2073611262835488442omplex(G,T) = groups2073611262835488442omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4567_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_rat(G,X4) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T) = groups2906978787729119204at_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4568_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_int(G,X4) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T) = groups3539618377306564664at_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4569_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_int] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_int(G,X4) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,T) = groups4538972089207619220nt_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4570_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
         => ( groups7754918857620584856omplex(G,T) = groups7754918857620584856omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4571_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),T) = aa_set_nat_nat(groups3542108847815614940at_nat(G),S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4572_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
           => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4573_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_complex(H,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4574_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4575_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4576_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4577_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4578_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,S3) = groups1300246762558778688al_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4579_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4580_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,S3) = groups4541462559716669496nt_nat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4581_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,S3) = groups1935376822645274424al_nat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4582_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
           => ( groups3049146728041665814omplex(G,T) = groups3049146728041665814omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4583_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_complex(G,X4) = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups5754745047067104278omplex(G,T) = groups5754745047067104278omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4584_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,T) = groups8778361861064173332t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4585_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups8097168146408367636l_real(G,T) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4586_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4587_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups3906332499630173760nt_rat(G,T) = groups3906332499630173760nt_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4588_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups1300246762558778688al_rat(G,T) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4589_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,T) = groups5058264527183730370ex_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4590_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,T) = groups4541462559716669496nt_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4591_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_nat(G,X4) = aa_real_nat(H,X4) ) )
           => ( groups1935376822645274424al_nat(G,T) = groups1935376822645274424al_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4592_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,B4))),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4593_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4594_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4595_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,B4))),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4596_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups2906978787729119204at_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,minus_minus_set_nat(A4,B4))),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4597_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups3539618377306564664at_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A4,B4))),groups3539618377306564664at_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4598_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_int] :
      ( ord_less_eq_set_int(B4,A4)
     => ( finite_finite_int(A4)
       => ( groups4538972089207619220nt_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A4,B4))),groups4538972089207619220nt_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4599_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_complex] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups7754918857620584856omplex(G,A4) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,minus_811609699411566653omplex(A4,B4))),groups7754918857620584856omplex(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4600_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),minus_minus_set_nat(A4,B4))),aa_set_nat_nat(groups3542108847815614940at_nat(G),B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4601_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups6591440286371151544t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,minus_minus_set_nat(A4,B4))),groups6591440286371151544t_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4602_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(A4)
     => ( ord_le211207098394363844omplex(B4,A4)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4603_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(A4)
     => ( ord_le211207098394363844omplex(B4,A4)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A4,B4)) = minus_minus_rat(groups5058264527183730370ex_rat(F,A4),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4604_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(A4)
     => ( ord_le211207098394363844omplex(B4,A4)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4605_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A4,B4)) = minus_minus_rat(groups2906978787729119204at_rat(F,A4),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4606_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A4)),groups3539618377306564664at_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4607_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_int] :
      ( finite_finite_int(A4)
     => ( ord_less_eq_set_int(B4,A4)
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A4)),groups4538972089207619220nt_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4608_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( finite3207457112153483333omplex(A4)
     => ( ord_le211207098394363844omplex(B4,A4)
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A4,B4)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A4)),groups7754918857620584856omplex(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4609_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( groups6591440286371151544t_real(F,minus_minus_set_nat(A4,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A4)),groups6591440286371151544t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4610_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_4611_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_4612_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_4613_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_4614_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,Mb)),aa_nat_set_nat(nat_set_decode,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_4615_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_real] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( ! [B3: int] :
              ( member_int(B3,minus_minus_set_int(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4616_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4617_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( ! [B3: complex] :
              ( member_complex(B3,minus_811609699411566653omplex(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4618_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_rat] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( ! [B3: int] :
              ( member_int(B3,minus_minus_set_int(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4619_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4620_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( ! [B3: complex] :
              ( member_complex(B3,minus_811609699411566653omplex(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4621_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( ! [B3: int] :
              ( member_int(B3,minus_minus_set_int(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4622_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4623_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( ! [B3: complex] :
              ( member_complex(B3,minus_811609699411566653omplex(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4624_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A4)),groups1932886352136224148al_int(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4625_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_4626_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_4627_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_4628_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] : aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_4629_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_4630_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_4631_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Mb,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_4632_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Mb),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_4633_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: fun_int_real] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( member_int(B,minus_minus_set_int(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4634_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X4: real] :
                  ( member_real(X4,B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4635_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( member_complex(B,minus_811609699411566653omplex(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X4: complex] :
                  ( member_complex(X4,B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4636_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: fun_int_rat] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( member_int(B,minus_minus_set_int(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4637_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X4: real] :
                  ( member_real(X4,B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4638_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( member_complex(B,minus_811609699411566653omplex(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X4: complex] :
                  ( member_complex(X4,B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4639_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: fun_int_nat] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( member_int(B,minus_minus_set_int(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,B))
           => ( ! [X4: int] :
                  ( member_int(X4,B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4640_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
           => ( ! [X4: real] :
                  ( member_real(X4,B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4641_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( member_complex(B,minus_811609699411566653omplex(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B))
           => ( ! [X4: complex] :
                  ( member_complex(X4,B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4642_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( member_real(B,minus_minus_set_real(B4,A4))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,B))
           => ( ! [X4: real] :
                  ( member_real(X4,B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X4)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A4)),groups1932886352136224148al_int(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4643_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => aa_nat_o(P,A3) )
     => ( ! [A3: nat,B3: $o] :
            ( aa_nat_o(P,A3)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat((B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat((B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_4644_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(divide_divide_int(A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => aa_int_o(P,A3) )
     => ( ! [A3: int,B3: $o] :
            ( aa_int_o(P,A3)
           => ( ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B3))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_4645_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( divide6298287555418463151nteger(A3,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
         => aa_Code_integer_o(P,A3) )
     => ( ! [A3: code_integer,B3: $o] :
            ( aa_Code_integer_o(P,A3)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_4646_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_4647_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_4648_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_4649_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_4650_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_4651_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_4652_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_4653_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_4654_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_4655_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_4656_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4657_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4658_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4659_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4660_crossproduct__eq,axiom,
    ! [W: real,Y: real,Xb: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4661_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,Xb: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4662_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,Xb: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4663_crossproduct__eq,axiom,
    ! [W: int,Y: int,Xb: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_4664_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_4665_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_4666_set__decode__def,axiom,
    ! [Xb: nat] : aa_nat_set_nat(nat_set_decode,Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gs,Xb)) ).

% set_decode_def
tff(fact_4667_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          zero_n2687167440665602831ol_nat(( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4668_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_int_int(divide_divide_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4669_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4670_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_4671_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_4672_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_4673_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),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,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_4674_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4675_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4676_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),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,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_4677_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)) ) )
           => ~ ! [Va3: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
                 => ( ( 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,Va3))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),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,Va3))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4678_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gt(L)),Qr) ).

% divmod_step_def
tff(fact_4679_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gu(L)),Qr) ).

% divmod_step_def
tff(fact_4680_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_gv(L),Qr) ).

% divmod_step_def
tff(fact_4681_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4682_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4683_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4684_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = uminus_uminus_rat(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4685_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4686_add_Oinverse__inverse,axiom,
    ! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).

% add.inverse_inverse
tff(fact_4687_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_4688_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_4689_add_Oinverse__inverse,axiom,
    ! [A: rat] : uminus_uminus_rat(uminus_uminus_rat(A)) = A ).

% add.inverse_inverse
tff(fact_4690_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4691_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_4692_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_4693_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_4694_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_4695_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_4696_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_4697_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_4698_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_4699_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_4700_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_4701_compl__le__compl__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),uminus5710092332889474511et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Y),Xb) ) ).

% compl_le_compl_iff
tff(fact_4702_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4703_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4704_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4705_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4706_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4707_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4708_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = uminus_uminus_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4709_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4710_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_4711_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_4712_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_4713_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_4714_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_4715_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4716_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4717_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4718_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = uminus_uminus_rat(A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4719_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4720_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4721_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4722_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4723_add_Oinverse__neutral,axiom,
    uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4724_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4725_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_4726_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_4727_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_4728_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_4729_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4730_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4731_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4732_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(Mb)) = uminus_uminus_rat(numeral_numeral_rat(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4733_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4734_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_left
tff(fact_4735_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_4736_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_4737_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(A)),B) = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_left
tff(fact_4738_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_4739_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).

% minus_mult_minus
tff(fact_4740_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_4741_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_4742_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(A)),uminus_uminus_rat(B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ).

% minus_mult_minus
tff(fact_4743_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_4744_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% mult_minus_right
tff(fact_4745_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_4746_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_4747_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(B)) = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_right
tff(fact_4748_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_4749_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_4750_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_4751_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_4752_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),B)) = B ).

% add_minus_cancel
tff(fact_4753_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_4754_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).

% minus_add_cancel
tff(fact_4755_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_4756_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_4757_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_4758_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_4759_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_4760_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_4761_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_4762_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),uminus_uminus_rat(B)) ).

% minus_add_distrib
tff(fact_4763_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_4764_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% minus_diff_eq
tff(fact_4765_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_4766_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_4767_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).

% minus_diff_eq
tff(fact_4768_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_4769_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(A),B) ).

% div_minus_minus
tff(fact_4770_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_4771_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_4772_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_4773_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_4774_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_4775_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_4776_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_4777_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_4778_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_4779_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_4780_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_4781_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_4782_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_4783_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_4784_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_4785_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_4786_abs__0,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_4787_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_4788_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_4789_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_4790_minus__dvd__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_4791_minus__dvd__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,uminus_uminus_rat(Xb)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_4792_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_4793_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_4794_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_4795_dvd__minus__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_complex_complex(uminus1482373934393186551omplex,Y))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_4796_dvd__minus__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),uminus_uminus_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_4797_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_4798_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).

% abs_numeral
tff(fact_4799_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,numeral_numeral_real(Nb)) = numeral_numeral_real(Nb) ).

% abs_numeral
tff(fact_4800_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_rat(numeral_numeral_rat(Nb)) = numeral_numeral_rat(Nb) ).

% abs_numeral
tff(fact_4801_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_4802_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_4803_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% abs_mult_self_eq
tff(fact_4804_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% abs_mult_self_eq
tff(fact_4805_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_4806_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_4807_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_4808_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_4809_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_4810_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_4811_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_4812_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_4813_abs__add__abs,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_add_abs
tff(fact_4814_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_4815_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_4816_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_4817_abs__divide,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ).

% abs_divide
tff(fact_4818_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_4819_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_4820_abs__minus__cancel,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus_cancel
tff(fact_4821_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_4822_abs__minus,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus
tff(fact_4823_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_4824_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_4825_abs__minus,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus
tff(fact_4826_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_4827_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B)) ).

% mod_minus_minus
tff(fact_4828_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_4829_dvd__abs__iff,axiom,
    ! [Mb: real,K2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),aa_real_real(abs_abs_real,K2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K2) ) ).

% dvd_abs_iff
tff(fact_4830_dvd__abs__iff,axiom,
    ! [Mb: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),aa_int_int(abs_abs_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K2) ) ).

% dvd_abs_iff
tff(fact_4831_dvd__abs__iff,axiom,
    ! [Mb: code_integer,K2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),abs_abs_Code_integer(K2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K2) ) ).

% dvd_abs_iff
tff(fact_4832_abs__dvd__iff,axiom,
    ! [Mb: real,K2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,Mb)),K2)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K2) ) ).

% abs_dvd_iff
tff(fact_4833_abs__dvd__iff,axiom,
    ! [Mb: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,Mb)),K2)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K2) ) ).

% abs_dvd_iff
tff(fact_4834_abs__dvd__iff,axiom,
    ! [Mb: code_integer,K2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(Mb)),K2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K2) ) ).

% abs_dvd_iff
tff(fact_4835_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_4836_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_4837_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_Code_integer(zero_n356916108424825756nteger((P))) = zero_n356916108424825756nteger((P)) ).

% abs_bool_eq
tff(fact_4838_case__prod__conv,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(F,A),B) ).

% case_prod_conv
tff(fact_4839_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_4840_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_4841_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_4842_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_4843_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_4844_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_4845_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_4846_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_4847_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_4848_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_4849_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_4850_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_4851_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_4852_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_4853_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_4854_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_4855_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_4856_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_4857_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_4858_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_4859_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_4860_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_4861_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_4862_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_4863_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_4864_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_4865_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_4866_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_4867_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_4868_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_4869_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_4870_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_4871_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_4872_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_4873_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_4874_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_4875_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_4876_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_4877_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_4878_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4879_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4880_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_4881_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_4882_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_4883_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4884_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4885_diff__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4886_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_4887_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4888_diff__0,axiom,
    ! [A: rat] : minus_minus_rat(zero_zero_rat,A) = uminus_uminus_rat(A) ).

% diff_0
tff(fact_4889_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4890_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_4891_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_4892_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_4893_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : minus_minus_rat(zero_zero_rat,B) = uminus_uminus_rat(B) ).

% verit_minus_simplify(3)
tff(fact_4894_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4895_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4896_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4897_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4898_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),uminus_uminus_rat(numeral_numeral_rat(Nb))) = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Mb)),numeral_numeral_rat(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4899_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4900_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_4901_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_4902_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_4903_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).

% mult_minus1_right
tff(fact_4904_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_4905_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_4906_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_4907_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_4908_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).

% mult_minus1
tff(fact_4909_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_4910_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_4911_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_4912_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_4913_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_4914_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_4915_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_4916_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_4917_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_4918_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_4919_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4920_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_nonneg
tff(fact_4921_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4922_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_4923_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_4924_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),abs_abs_rat(A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_4925_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_4926_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).

% diff_minus_eq_add
tff(fact_4927_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_4928_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_4929_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,uminus_uminus_rat(B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_4930_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_4931_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).

% uminus_add_conv_diff
tff(fact_4932_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_4933_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_4934_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),B) = minus_minus_rat(B,A) ).

% uminus_add_conv_diff
tff(fact_4935_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_4936_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_4937_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_4938_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_4939_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_4940_divide__minus1,axiom,
    ! [Xb: rat] : divide_divide_rat(Xb,uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Xb) ).

% divide_minus1
tff(fact_4941_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(Nb) ).

% abs_neg_numeral
tff(fact_4942_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_4943_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_rat(uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(Nb) ).

% abs_neg_numeral
tff(fact_4944_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).

% abs_neg_numeral
tff(fact_4945_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_4946_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_4947_abs__neg__one,axiom,
    abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_4948_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_4949_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_4950_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_mod_self1
tff(fact_4951_abs__power__minus,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% abs_power_minus
tff(fact_4952_abs__power__minus,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) ).

% abs_power_minus
tff(fact_4953_abs__power__minus,axiom,
    ! [A: rat,Nb: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb)) = abs_abs_rat(aa_nat_rat(power_power_rat(A),Nb)) ).

% abs_power_minus
tff(fact_4954_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% abs_power_minus
tff(fact_4955_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_4956_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_4957_real__add__minus__iff,axiom,
    ! [Xb: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xb = A ) ) ).

% real_add_minus_iff
tff(fact_4958_sum__abs,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A4))),groups4538972089207619220nt_int(aTP_Lamm_gw(F),A4)) ).

% sum_abs
tff(fact_4959_sum__abs,axiom,
    ! [F: fun_nat_real,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,groups6591440286371151544t_real(F,A4))),groups6591440286371151544t_real(aTP_Lamm_gx(F),A4)) ).

% sum_abs
tff(fact_4960_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K2))) ).

% dbl_simps(1)
tff(fact_4961_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_simps(1)
tff(fact_4962_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2))) ).

% dbl_simps(1)
tff(fact_4963_dbl__simps_I1_J,axiom,
    ! [K2: num] : neg_numeral_dbl_rat(uminus_uminus_rat(numeral_numeral_rat(K2))) = uminus_uminus_rat(neg_numeral_dbl_rat(numeral_numeral_rat(K2))) ).

% dbl_simps(1)
tff(fact_4964_dbl__simps_I1_J,axiom,
    ! [K2: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K2))) ).

% dbl_simps(1)
tff(fact_4965_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_4966_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_4967_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_4968_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4969_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_4970_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_4971_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_4972_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_4973_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4974_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_4975_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_4976_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_4977_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_4978_diff__numeral__special_I12_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4979_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_4980_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4981_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4982_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4983_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( uminus_uminus_rat(one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4984_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4985_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4986_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4987_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4988_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(Nb)) = uminus_uminus_rat(one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4989_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4990_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_4991_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A,abs_abs_rat(B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_4992_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_4993_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,abs_abs_rat(B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_4994_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_4995_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_4996_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_nonpos
tff(fact_4997_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_4998_minus__one__mult__self,axiom,
    ! [Nb: 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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_4999_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_5000_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_5001_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_5002_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_5003_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),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)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5004_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5005_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5006_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5007_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5008_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_5009_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5010_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,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(4)
tff(fact_5011_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_5012_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),uminus_uminus_rat(numeral_numeral_rat(V))),uminus_uminus_rat(numeral_numeral_rat(V)),uminus_uminus_rat(numeral_numeral_rat(U))) ).

% max_number_of(4)
tff(fact_5013_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_5014_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,numeral_numeral_real(U))),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)),numeral_numeral_real(V),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(3)
tff(fact_5015_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_5016_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),numeral_numeral_rat(V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),numeral_numeral_rat(V)),numeral_numeral_rat(V),uminus_uminus_rat(numeral_numeral_rat(U))) ).

% max_number_of(3)
tff(fact_5017_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_5018_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),numeral_numeral_real(U)) ).

% max_number_of(2)
tff(fact_5019_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_5020_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(numeral_numeral_rat(U),uminus_uminus_rat(numeral_numeral_rat(V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))),uminus_uminus_rat(numeral_numeral_rat(V)),numeral_numeral_rat(U)) ).

% max_number_of(2)
tff(fact_5021_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_5022_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5023_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5024_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5025_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5026_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_5027_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5028_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5029_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5030_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_minus_rat(numeral_numeral_rat(Mb),uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5031_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5032_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),numeral_numeral_real(Nb)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5033_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5034_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5035_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(Mb)),numeral_numeral_rat(Nb)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5036_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5037_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5038_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5039_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5040_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5041_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5042_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),numeral_numeral_real(Nb)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5043_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5044_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5045_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),numeral_numeral_rat(Nb)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5046_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5047_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5048_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5049_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5050_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(Mb)),uminus_uminus_rat(numeral_numeral_rat(Nb))) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5051_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5052_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5053_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5054_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5055_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5056_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5057_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5058_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5059_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5060_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5061_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5062_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5063_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5064_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5065_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5066_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5067_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_5068_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5069_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5070_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),uminus_uminus_rat(numeral_numeral_rat(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5071_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5072_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5073_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5074_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),uminus_uminus_rat(numeral_numeral_rat(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5075_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5076_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_gw(F),A4)) ).

% sum_abs_ge_zero
tff(fact_5077_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A4: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_gx(F),A4)) ).

% sum_abs_ge_zero
tff(fact_5078_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5079_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5080_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5081_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5082_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5083_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5084_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5085_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5086_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5087_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5088_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) = A )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5089_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5090_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5091_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5092_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5093_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5094_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),uminus_uminus_rat(one_one_rat))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5095_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5096_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5097_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5098_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5099_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5100_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5101_power2__minus,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5102_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5103_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5104_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5105_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5106_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5107_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5108_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5109_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5110_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5111_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5112_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5113_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5114_power2__abs,axiom,
    ! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5115_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5116_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5117_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5118_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5119_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_5120_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A4: set_nat] :
      groups2073611262835488442omplex(aTP_Lamm_gy(C),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_5121_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A4: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_gz(C),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_5122_sum__zero__power,axiom,
    ! [C: fun_nat_real,A4: set_nat] :
      groups6591440286371151544t_real(aTP_Lamm_ha(C),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_5123_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5124_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_5125_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_5126_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5127_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5128_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5129_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_5130_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_5131_diff__numeral__special_I11_J,axiom,
    minus_minus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5132_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5133_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5134_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_5135_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_5136_diff__numeral__special_I10_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5137_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5138_minus__1__div__2__eq,axiom,
    aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_5139_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_5140_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_5141_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_5142_minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_5143_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_5144_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5145_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5146_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5147_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5148_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5149_power__minus__odd,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5150_power__minus__odd,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5151_power__minus__odd,axiom,
    ! [Nb: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5152_power__minus__odd,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5153_power__minus__odd,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5154_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5155_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5156_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5157_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5158_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5159_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5160_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5161_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5162_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5163_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5164_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5165_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_minus_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(Nb))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5166_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5167_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5168_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5169_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5170_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(Mb)),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5171_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5172_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5173_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5174_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_5175_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_5176_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5177_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5178_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A4: set_nat] :
      groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_hb(C),D),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        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_5179_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A4: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hc(C),D),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        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_5180_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A4: set_nat] :
      groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_hd(C),D),A4) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
        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_5181_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_5182_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_5183_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_5184_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_5185_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_5186_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_5187_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_5188_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_5189_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb) = uminus_uminus_rat(one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_5190_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_5191_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_5192_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_5193_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_5194_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_5195_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_5196_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5197_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5198_signed__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,K2)) ).

% signed_take_bit_minus
tff(fact_5199_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_5200_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_5201_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B) ) ) ).

% abs_leI
tff(fact_5202_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_5203_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_5204_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_5205_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B) ) ).

% abs_le_D2
tff(fact_5206_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_5207_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_5208_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_5209_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B) ) ) ).

% abs_le_iff
tff(fact_5210_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_5211_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_5212_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_5213_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),abs_abs_rat(A)) ).

% abs_ge_minus_self
tff(fact_5214_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_5215_prod_Ocase__distrib,axiom,
    ! [H: fun_nat_nat,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat_nat(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_he(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5216_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_hf(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5217_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_hg(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5218_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_hh(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5219_prod_Ocase__distrib,axiom,
    ! [H: fun_na3970208099681795815_nat_o,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat7695548938202568488_nat_o(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun2100511394839469148_nat_o(aTP_Lamm_hi(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5220_prod_Ocase__distrib,axiom,
    ! [H: fun_fu1115759055147092071_o_nat,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun1432809138467554728_o_nat(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun3680403850923830236at_nat(aTP_Lamm_hj(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5221_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_hk(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5222_prod_Ocase__distrib,axiom,
    ! [H: fun_na8713714802964096592at_nat,F: fun_nat_fun_nat_nat,Prod: product_prod_nat_nat] : aa_nat7997280740862404369at_nat(H,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),Prod)) = aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aa_fun2484538639213223237at_nat(aTP_Lamm_hl(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5223_prod_Ocase__distrib,axiom,
    ! [H: fun_fu1413608398870722896at_nat,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] : aa_fun7582795104022253201at_nat(H,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),Prod)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aa_fun8817627472026143685at_nat(aTP_Lamm_hm(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5224_prod_Ocase__distrib,axiom,
    ! [H: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_hn(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5225_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_5226_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_5227_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_5228_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( uminus_uminus_rat(B) = A ) ) ).

% minus_equation_iff
tff(fact_5229_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_5230_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_5231_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_5232_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_5233_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% equation_minus_iff
tff(fact_5234_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_5235_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_5236_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_5237_abs__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( abs_abs_rat(Xb) = abs_abs_rat(Y) )
    <=> ( ( Xb = Y )
        | ( Xb = uminus_uminus_rat(Y) ) ) ) ).

% abs_eq_iff
tff(fact_5238_abs__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(Xb) = abs_abs_Code_integer(Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5239_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_5240_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_5241_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),B) ) ) ).

% abs_less_iff
tff(fact_5242_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_5243_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_5244_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_5245_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(abs_abs_rat(A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_5246_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_5247_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5248_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5249_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat(A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = uminus_uminus_rat(B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5250_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5251_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5252_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5253_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = abs_abs_rat(B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = uminus_uminus_rat(A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5254_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5255_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_5256_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_5257_abs__if,axiom,
    ! [A: rat] :
      abs_abs_rat(A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),uminus_uminus_rat(A),A) ).

% abs_if
tff(fact_5258_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_5259_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_5260_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_5261_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_neg
tff(fact_5262_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_5263_abs__if__raw,axiom,
    ! [X: real] :
      aa_real_real(abs_abs_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real),aa_real_real(uminus_uminus_real,X),X) ).

% abs_if_raw
tff(fact_5264_abs__if__raw,axiom,
    ! [X: int] :
      aa_int_int(abs_abs_int,X) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int),aa_int_int(uminus_uminus_int,X),X) ).

% abs_if_raw
tff(fact_5265_abs__if__raw,axiom,
    ! [X: rat] :
      abs_abs_rat(X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat),uminus_uminus_rat(X),X) ).

% abs_if_raw
tff(fact_5266_abs__if__raw,axiom,
    ! [X: code_integer] :
      abs_abs_Code_integer(X) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),X) ).

% abs_if_raw
tff(fact_5267_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_5268_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_5269_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_5270_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_5271_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_5272_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_5273_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_5274_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),abs_abs_rat(A)) ).

% abs_ge_self
tff(fact_5275_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_5276_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5277_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_5278_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_5279_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_5280_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_5281_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_5282_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_5283_abs__mult,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_mult
tff(fact_5284_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_5285_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_5286_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_5287_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_5288_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_5289_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_5290_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% abs_minus_commute
tff(fact_5291_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(minus_minus_rat(A,B)) = abs_abs_rat(minus_minus_rat(B,A)) ).

% abs_minus_commute
tff(fact_5292_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_5293_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_5294_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).

% power_abs
tff(fact_5295_power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) ).

% power_abs
tff(fact_5296_power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) ).

% power_abs
tff(fact_5297_dvd__if__abs__eq,axiom,
    ! [L: real,K2: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K2) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),K2) ) ).

% dvd_if_abs_eq
tff(fact_5298_dvd__if__abs__eq,axiom,
    ! [L: int,K2: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K2) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2) ) ).

% dvd_if_abs_eq
tff(fact_5299_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K2: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K2) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K2) ) ).

% dvd_if_abs_eq
tff(fact_5300_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_5301_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_5302_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),uminus_uminus_rat(A)) ) ).

% le_minus_iff
tff(fact_5303_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_5304_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_5305_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_5306_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),A) ) ).

% minus_le_iff
tff(fact_5307_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_5308_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_5309_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_5310_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ) ).

% le_imp_neg_le
tff(fact_5311_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_5312_compl__le__swap2,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),Xb)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),Y) ) ).

% compl_le_swap2
tff(fact_5313_compl__le__swap1,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(Xb))
     => aa_set_nat_o(ord_less_eq_set_nat(Xb),uminus5710092332889474511et_nat(Y)) ) ).

% compl_le_swap1
tff(fact_5314_compl__mono,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(Xb)) ) ).

% compl_mono
tff(fact_5315_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_5316_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_5317_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),uminus_uminus_rat(A)) ) ).

% less_minus_iff
tff(fact_5318_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_5319_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_5320_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_5321_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),A) ) ).

% minus_less_iff
tff(fact_5322_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_5323_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5324_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5325_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5326_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5327_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb)) != numeral_numeral_real(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5328_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5329_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5330_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : uminus_uminus_rat(numeral_numeral_rat(Mb)) != numeral_numeral_rat(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5331_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) != numera6620942414471956472nteger(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5332_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numeral_numeral_real(Mb) != aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5333_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,Mb) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5334_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6690914467698888265omplex(Mb) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5335_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numeral_numeral_rat(Mb) != uminus_uminus_rat(numeral_numeral_rat(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5336_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6620942414471956472nteger(Mb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5337_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_5338_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_5339_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_5340_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A = B )
        | ( A = uminus_uminus_rat(B) ) ) ) ).

% square_eq_iff
tff(fact_5341_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_5342_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_5343_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_5344_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_5345_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(B)) ).

% minus_mult_commute
tff(fact_5346_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_5347_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_5348_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_5349_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_5350_one__neq__neg__one,axiom,
    one_one_rat != uminus_uminus_rat(one_one_rat) ).

% one_neq_neg_one
tff(fact_5351_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_5352_group__cancel_Oneg1,axiom,
    ! [A4: real,K2: real,A: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),A) )
     => ( aa_real_real(uminus_uminus_real,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K2)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_5353_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(uminus_uminus_int,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_5354_group__cancel_Oneg1,axiom,
    ! [A4: complex,K2: complex,A: complex] :
      ( ( A4 = aa_complex_complex(plus_plus_complex(K2),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_5355_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),A) )
     => ( uminus_uminus_rat(A4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(K2)),uminus_uminus_rat(A)) ) ) ).

% group_cancel.neg1
tff(fact_5356_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4 = plus_p5714425477246183910nteger(K2,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_5357_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_5358_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_5359_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_5360_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ).

% add.inverse_distrib_swap
tff(fact_5361_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_5362_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_5363_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_5364_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_5365_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ).

% is_num_normalize(8)
tff(fact_5366_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_5367_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% minus_diff_minus
tff(fact_5368_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ).

% minus_diff_minus
tff(fact_5369_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).

% minus_diff_minus
tff(fact_5370_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(minus_minus_rat(A,B)) ).

% minus_diff_minus
tff(fact_5371_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_5372_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_5373_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_5374_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_5375_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : minus_minus_rat(uminus_uminus_rat(B),A) = minus_minus_rat(uminus_uminus_rat(A),B) ).

% minus_diff_commute
tff(fact_5376_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_5377_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_5378_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% div_minus_right
tff(fact_5379_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_5380_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_5381_minus__divide__right,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ).

% minus_divide_right
tff(fact_5382_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).

% minus_divide_divide
tff(fact_5383_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_5384_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ).

% minus_divide_divide
tff(fact_5385_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_5386_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_5387_minus__divide__left,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(uminus_uminus_rat(A),B) ).

% minus_divide_left
tff(fact_5388_old_Oprod_Ocase,axiom,
    ! [F: fun_nat_fun_nat_nat,X1: nat,X2: nat] : aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat_nat(aa_nat_fun_nat_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5389_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5390_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X2) ).

% old.prod.case
tff(fact_5391_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X2) ).

% old.prod.case
tff(fact_5392_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X2: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X2) ) ).

% old.prod.case
tff(fact_5393_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_5394_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).

% mod_minus_right
tff(fact_5395_mod__minus__cong,axiom,
    ! [A: int,B: int,A2: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = aa_int_int(modulo_modulo_int(A2),B) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A2)),B) ) ) ).

% mod_minus_cong
tff(fact_5396_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A2,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),B) ) ) ).

% mod_minus_cong
tff(fact_5397_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A),B))),B) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_5398_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% mod_minus_eq
tff(fact_5399_cond__case__prod__eta,axiom,
    ! [F: fun_nat_fun_nat_nat,G: fun_Pr2602117396471556196at_nat] :
      ( ! [X4: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(F,X4),Y3) = aa_Pro4243173770044397973at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))
     => ( produc6842872674320459806at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5400_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X4: nat,Y3: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y3) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5401_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X4: nat,Y3: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y3) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3))
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5402_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X4: int,Y3: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y3) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5403_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X4: int,Y3: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X4),Y3)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5404_case__prod__eta,axiom,
    ! [F: fun_Pr2602117396471556196at_nat] : produc6842872674320459806at_nat(aTP_Lamm_ho(F)) = F ).

% case_prod_eta
tff(fact_5405_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_hp(F)) = F ).

% case_prod_eta
tff(fact_5406_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_hq(F)) = F ).

% case_prod_eta
tff(fact_5407_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_hr(F)) = F ).

% case_prod_eta
tff(fact_5408_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_hs(F)) = F ).

% case_prod_eta
tff(fact_5409_case__prodE2,axiom,
    ! [Q: fun_nat_o,P: fun_nat_fun_nat_nat,Z: product_prod_nat_nat] :
      ( aa_nat_o(Q,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(P),Z))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ aa_nat_o(Q,aa_nat_nat(aa_nat_fun_nat_nat(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_5410_case__prodE2,axiom,
    ! [Q: fun_fu8391177546919232046_nat_o,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_5411_case__prodE2,axiom,
    ! [Q: fun_fu6891447142836604759at_o_o,P: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(P),Z))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_5412_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
     => ~ ! [X4: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_5413_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
     => ~ ! [X4: int,Y3: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_5414_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_5415_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_5416_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A)) ).

% abs_ge_zero
tff(fact_5417_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_5418_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_5419_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_5420_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_5421_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_5422_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_5423_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_5424_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_pos
tff(fact_5425_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_5426_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_5427_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_5428_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq
tff(fact_5429_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_5430_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_5431_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_5432_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D)) ) ) ).

% abs_mult_less
tff(fact_5433_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_5434_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_5435_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5436_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_5437_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5438_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_5439_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq3
tff(fact_5440_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B)))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq3
tff(fact_5441_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq3
tff(fact_5442_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_5443_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq2
tff(fact_5444_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(A,B))) ).

% abs_triangle_ineq2
tff(fact_5445_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq2
tff(fact_5446_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_5447_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ) ) ).

% nonzero_abs_divide
tff(fact_5448_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X4)),A4)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X4)) = aa_nat_nat(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_5449_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X4)),A4)
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( groups6591440286371151544t_real(F,A4) = groups6591440286371151544t_real(G,A4) ) ) ) ).

% sum_cong_Suc
tff(fact_5450_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),numeral_numeral_real(Nb)) ).

% neg_numeral_le_numeral
tff(fact_5451_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_5452_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),numeral_numeral_rat(Nb)) ).

% neg_numeral_le_numeral
tff(fact_5453_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5454_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5455_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5456_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Mb)),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5457_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5458_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5459_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5460_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5461_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != uminus_uminus_rat(numeral_numeral_rat(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5462_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5463_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5464_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5465_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Mb)),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5466_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5467_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),numeral_numeral_real(Nb)) ).

% neg_numeral_less_numeral
tff(fact_5468_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5469_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),numeral_numeral_rat(Nb)) ).

% neg_numeral_less_numeral
tff(fact_5470_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_less_numeral
tff(fact_5471_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_5472_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_5473_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_5474_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_5475_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_5476_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_5477_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_5478_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_5479_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5480_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5481_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5482_zero__neq__neg__one,axiom,
    zero_zero_rat != uminus_uminus_rat(one_one_rat) ).

% zero_neq_neg_one
tff(fact_5483_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5484_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5485_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5486_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5487_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5488_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5489_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5490_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5491_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5492_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5493_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5494_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_5495_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_5496_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_5497_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( uminus_uminus_rat(A) = B ) ) ).

% add.inverse_unique
tff(fact_5498_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_5499_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_5500_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_5501_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_5502_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5503_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_5504_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_5505_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_5506_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_5507_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% add_eq_0_iff
tff(fact_5508_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_5509_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_5510_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_5511_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_5512_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_5513_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_5514_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_5515_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_5516_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_5517_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% numeral_times_minus_swap
tff(fact_5518_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_5519_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_5520_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),uminus_uminus_rat(Xb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),uminus_uminus_rat(numeral_numeral_rat(W))) ).

% numeral_times_minus_swap
tff(fact_5521_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_5522_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5523_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5524_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5525_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5526_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5527_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5528_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb)) ).

% one_neq_neg_numeral
tff(fact_5529_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_5530_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_5531_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(Nb)) ).

% one_neq_neg_numeral
tff(fact_5532_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% one_neq_neg_numeral
tff(fact_5533_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numeral_numeral_real(Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5534_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_5535_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5536_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numeral_numeral_rat(Nb) != uminus_uminus_rat(one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5537_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6620942414471956472nteger(Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5538_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_5539_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_5540_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_5541_square__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb) = one_one_rat )
    <=> ( ( Xb = one_one_rat )
        | ( Xb = uminus_uminus_rat(one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_5542_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_5543_group__cancel_Osub2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K2),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5544_group__cancel_Osub2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5545_group__cancel_Osub2,axiom,
    ! [B4: complex,K2: complex,B: complex,A: complex] :
      ( ( B4 = aa_complex_complex(plus_plus_complex(K2),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5546_group__cancel_Osub2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K2),B) )
     => ( minus_minus_rat(A,B4) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(K2)),minus_minus_rat(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5547_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B4 = plus_p5714425477246183910nteger(K2,B) )
     => ( minus_8373710615458151222nteger(A,B4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5548_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_5549_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_5550_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_5551_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(B)) ).

% diff_conv_add_uminus
tff(fact_5552_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_5553_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5554_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5555_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5556_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : minus_minus_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5557_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5558_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5559_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5560_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5561_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(uminus_uminus_rat(A),B) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_neg_div
tff(fact_5562_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_5563_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5564_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5565_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5566_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( divide_divide_rat(A,uminus_uminus_rat(B)) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).

% dvd_div_neg
tff(fact_5567_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_5568_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_5569_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_5570_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
     => ( ( Mb = one_one_int )
        | ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5571_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5572_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_5573_minus__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_5574_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ht(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5575_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_hu(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5576_sum__subtractf__nat,axiom,
    ! [A4: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X4)),aa_complex_nat(F,X4)) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_hv(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5577_sum__subtractf__nat,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X4)),aa_Pro4243173770044397973at_nat(F,X4)) )
     => ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_hw(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A4)),groups977919841031483927at_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5578_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hx(G),F)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_5579_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hy(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5580_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups6591440286371151544t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_5581_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ia(G),K2)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5582_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K2: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ib(G),K2),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_5583_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_5584_dense__eq0__I,axiom,
    ! [Xb: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),E2) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5585_dense__eq0__I,axiom,
    ! [Xb: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xb)),E2) )
     => ( Xb = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5586_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_5587_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_5588_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% abs_eq_mult
tff(fact_5589_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_5590_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,abs_abs_Code_integer(Y)),Xb) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5591_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_5592_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(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(Y)),Xb) = abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5593_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_5594_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_5595_abs__div__pos,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( divide_divide_rat(abs_abs_rat(Xb),Y) = abs_abs_rat(divide_divide_rat(Xb,Y)) ) ) ).

% abs_div_pos
tff(fact_5596_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_5597_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_5598_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),Nb)) ).

% zero_le_power_abs
tff(fact_5599_zero__le__power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb)) ).

% zero_le_power_abs
tff(fact_5600_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5601_abs__diff__le__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5602_abs__diff__le__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Xb,A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A,R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5603_abs__diff__le__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5604_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_5605_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_5606_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(A,B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq4
tff(fact_5607_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_5608_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_5609_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5610_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(minus_minus_rat(A,C))),abs_abs_rat(minus_minus_rat(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_5611_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5612_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5613_abs__diff__less__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5614_abs__diff__less__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xb,A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A,R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5615_abs__diff__less__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5616_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( ( groups5693394587270226106ex_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( member_complex(X3,A4)
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: complex] :
                ( member_complex(Xa2,A4)
               => ( ( X3 != Xa2 )
                 => ( aa_complex_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5617_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( ( X3 != Xa2 )
                 => ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5618_sum__SucD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,Nb) )
     => ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) ) ) ).

% sum_SucD
tff(fact_5619_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( ( groups5693394587270226106ex_nat(F,A4) = one_one_nat )
      <=> ? [X3: complex] :
            ( member_complex(X3,A4)
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa2: complex] :
                ( member_complex(Xa2,A4)
               => ( ( X3 != Xa2 )
                 => ( aa_complex_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5620_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = one_one_nat )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( ( X3 != Xa2 )
                 => ( aa_nat_nat(F,Xa2) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5621_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_5622_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_5623_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_5624_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_5625_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5626_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5627_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5628_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5629_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5630_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5631_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5632_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5633_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_5634_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_5635_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_5636_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_5637_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_5638_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_5639_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_5640_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_5641_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_5642_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_5643_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_5644_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_5645_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_5646_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_5647_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_5648_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_5649_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_5650_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_5651_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_5652_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_5653_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_5654_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_5655_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_5656_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_5657_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(Mb)) ).

% neg_one_le_numeral
tff(fact_5658_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_le_numeral
tff(fact_5659_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(Mb)) ).

% neg_one_le_numeral
tff(fact_5660_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_le_numeral
tff(fact_5661_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_5662_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_5663_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),uminus_uminus_rat(one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_5664_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_5665_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_5666_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_5667_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Mb)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_5668_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_5669_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))) ).

% not_one_le_neg_numeral
tff(fact_5670_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_le_neg_numeral
tff(fact_5671_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(Mb))) ).

% not_one_le_neg_numeral
tff(fact_5672_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_le_neg_numeral
tff(fact_5673_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5674_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5675_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5676_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5677_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))) ).

% not_one_less_neg_numeral
tff(fact_5678_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_less_neg_numeral
tff(fact_5679_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(Mb))) ).

% not_one_less_neg_numeral
tff(fact_5680_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_less_neg_numeral
tff(fact_5681_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_5682_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_5683_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Mb)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_5684_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_5685_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(Mb)) ).

% neg_one_less_numeral
tff(fact_5686_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_less_numeral
tff(fact_5687_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(Mb)) ).

% neg_one_less_numeral
tff(fact_5688_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_less_numeral
tff(fact_5689_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Mb))),one_one_real) ).

% neg_numeral_less_one
tff(fact_5690_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_less_one
tff(fact_5691_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(Mb))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_5692_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_5693_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_5694_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_5695_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = uminus_uminus_rat(divide_divide_rat(B,C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = uminus_uminus_rat(B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_5696_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_5697_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_5698_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( uminus_uminus_rat(divide_divide_rat(B,C)) = A )
    <=> $ite(C != zero_zero_rat,uminus_uminus_rat(B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_5699_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5700_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5701_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat(divide_divide_rat(A,B)) = C )
      <=> ( uminus_uminus_rat(A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5702_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5703_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5704_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = uminus_uminus_rat(divide_divide_rat(A,B)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) = uminus_uminus_rat(A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5705_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_5706_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_5707_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_5708_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(2)
tff(fact_5709_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_5710_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_5711_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_5712_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_5713_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(one))),B) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(1)
tff(fact_5714_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_5715_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5716_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5717_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat(A,B) = uminus_uminus_rat(one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = uminus_uminus_rat(B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5718_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,numeral_numeral_real(one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_5719_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_5720_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_5721_uminus__numeral__One,axiom,
    uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).

% uminus_numeral_One
tff(fact_5722_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_5723_power__minus,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_minus
tff(fact_5724_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_minus
tff(fact_5725_power__minus,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_minus
tff(fact_5726_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_5727_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_minus
tff(fact_5728_power__minus__Bit0,axiom,
    ! [Xb: real,K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5729_power__minus__Bit0,axiom,
    ! [Xb: int,K2: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5730_power__minus__Bit0,axiom,
    ! [Xb: complex,K2: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5731_power__minus__Bit0,axiom,
    ! [Xb: rat,K2: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5732_power__minus__Bit0,axiom,
    ! [Xb: code_integer,K2: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5733_lemma__interval__lt,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_5734_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_5735_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_5736_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_5737_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_5738_sum__power__add,axiom,
    ! [Xb: complex,Mb: nat,I6: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ic(Xb),Mb),I6) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),groups2073611262835488442omplex(power_power_complex(Xb),I6)) ).

% sum_power_add
tff(fact_5739_sum__power__add,axiom,
    ! [Xb: rat,Mb: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_id(Xb),Mb),I6) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),I6)) ).

% sum_power_add
tff(fact_5740_sum__power__add,axiom,
    ! [Xb: int,Mb: nat,I6: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ie(Xb),Mb),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),groups3539618377306564664at_int(power_power_int(Xb),I6)) ).

% sum_power_add
tff(fact_5741_sum__power__add,axiom,
    ! [Xb: real,Mb: nat,I6: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_if(Xb),Mb),I6) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),I6)) ).

% sum_power_add
tff(fact_5742_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_5743_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(modulo_modulo_int(A),B))) ).

% zmod_zminus1_eq_if
tff(fact_5744_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ig(G),Nb),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_5745_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ih(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_5746_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_ak,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cq,Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_5747_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_ak,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dn(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_5748_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xb))) ).

% abs_add_one_gt_zero
tff(fact_5749_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_5750_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(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),abs_abs_rat(Xb))) ).

% abs_add_one_gt_zero
tff(fact_5751_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_5752_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(B4,A4)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_5753_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_5754_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K2: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5755_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K2: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5756_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K2: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5757_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K2: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5758_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K2)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_5759_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5760_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5761_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5762_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_5763_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5764_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5765_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5766_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_5767_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5768_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5769_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5770_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_5771_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5772_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5773_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5774_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5775_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5776_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5777_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5778_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5779_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5780_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5781_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_5782_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_5783_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5784_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5785_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B,C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C),uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5786_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) = B,aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5787_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5788_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C) = B,uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5789_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_5790_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5791_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(divide_divide_rat(Xb,Z))),Y) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5792_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5793_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5794_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(divide_divide_rat(A,Z))),B) = $ite(Z = zero_zero_rat,B,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5795_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_5796_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5797_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(Xb,Z)),Y) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5798_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5799_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5800_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(divide_divide_rat(A,Z),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),divide_divide_rat(minus_minus_rat(A,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5801_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5802_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5803_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),divide_divide_rat(minus_minus_rat(uminus_uminus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z)),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5804_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_5805_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_5806_lemma__interval,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_5807_power2__eq__iff,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))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5808_power2__eq__iff,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))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5809_power2__eq__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5810_power2__eq__iff,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))) )
    <=> ( ( Xb = Y )
        | ( Xb = uminus_uminus_rat(Y) ) ) ) ).

% power2_eq_iff
tff(fact_5811_power2__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_5812_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_5813_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5814_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5815_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hy(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5816_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),groups6591440286371151544t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_5817_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_ik(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb)),aa_nat_rat(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5818_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_il(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5819_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups6591440286371151544t_real(aTP_Lamm_im(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_5820_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_5821_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),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_5822_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_5823_abs__le__square__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xb)),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_5824_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_5825_abs__square__eq__1,axiom,
    ! [Xb: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xb) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_5826_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 )
    <=> ( abs_abs_rat(Xb) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_5827_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_5828_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_5829_power__even__abs,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_even_abs
tff(fact_5830_power__even__abs,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_even_abs
tff(fact_5831_power__even__abs,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_even_abs
tff(fact_5832_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5833_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5834_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5835_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_5836_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5837_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5838_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5839_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5840_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5841_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5842_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5843_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5844_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_5845_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_5846_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_5847_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_5848_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5849_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5850_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5851_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5852_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_5853_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_5854_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_5855_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = uminus_uminus_rat(one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_5856_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_5857_uminus__power__if,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ).

% uminus_power_if
tff(fact_5858_uminus__power__if,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(power_power_int(A),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb))) ).

% uminus_power_if
tff(fact_5859_uminus__power__if,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(power_power_complex(A),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb))) ).

% uminus_power_if
tff(fact_5860_uminus__power__if,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A),Nb),uminus_uminus_rat(aa_nat_rat(power_power_rat(A),Nb))) ).

% uminus_power_if
tff(fact_5861_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).

% uminus_power_if
tff(fact_5862_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5863_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5864_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5865_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5866_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5867_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_5868_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_5869_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K2)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5870_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5871_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5872_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K2)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_5873_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5874_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_5875_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(modulo_modulo_int(A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_5876_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xb: code_integer] :
      ( ! [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X4),aa_nat_Code_integer(power_8256067586552552935nteger(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5877_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
         => aa_real_o(aa_real_fun_real_o(P,X4),aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => 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_5878_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xb: rat] :
      ( ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X4)
         => aa_rat_o(aa_rat_fun_rat_o(P,X4),aa_nat_rat(power_power_rat(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,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_5879_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: int] :
      ( ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_int_o(aa_int_fun_int_o(P,X4),aa_nat_int(power_power_int(X4),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_5880_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,abs_abs_Code_integer(Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5881_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_5882_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,abs_abs_rat(Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5883_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_5884_abs__square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_5885_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_5886_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,abs_abs_rat(Xb)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_5887_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_5888_abs__square__less__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_5889_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_5890_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,abs_abs_rat(Xb)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_5891_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_5892_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_5893_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_5894_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A)),abs_abs_rat(B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_5895_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_even
tff(fact_5896_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q3),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q3)),one_one_int))),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_5897_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5898_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5899_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5900_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5901_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xb,I2)) )
     => ( ( groups7501900531339628137nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_in(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5902_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xb,I2)) )
     => ( ( groups7873554091576472773nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_io(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5903_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xb,I2)) )
     => ( ( groups7713935264441627589nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_ip(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5904_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(Xb,I2)) )
     => ( ( groups6621422865394947399nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I2),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_iq(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5905_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xb,I2)) )
     => ( ( groups8778361861064173332t_real(Xb,I6) = one_one_real )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ir(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5906_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xb,I2)) )
     => ( ( groups8097168146408367636l_real(Xb,I6) = one_one_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_is(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5907_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Xb,I2)) )
     => ( ( groups5808333547571424918x_real(Xb,I6) = one_one_real )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_it(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5908_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xb,I2)) )
     => ( ( groups2906978787729119204at_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_iu(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5909_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_rat,A: fun_int_rat,B: rat,Delta: rat] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Xb,I2)) )
     => ( ( groups3906332499630173760nt_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_iv(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5910_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xb,I2)) )
     => ( ( groups1300246762558778688al_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_real_rat(A,I2),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_iw(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_5911_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),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,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_5912_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_5913_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_5914_square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_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_5915_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_5916_minus__power__mult__self,axiom,
    ! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5917_minus__power__mult__self,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5918_minus__power__mult__self,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5919_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb)),aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5920_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5921_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_5922_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_5923_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_5924_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,uminus_uminus_rat(one_one_rat)) ).

% minus_one_power_iff
tff(fact_5925_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_5926_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(aTP_Lamm_ix(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_complex_complex(minus_minus_complex(aa_nat_complex(F,Mb)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_5927_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_iy(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),minus_minus_rat(aa_nat_rat(F,Mb),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_5928_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_iz(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_5929_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(aTP_Lamm_ja(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_5930_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_jb(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = minus_minus_rat(aa_nat_rat(F,Nb),aa_nat_rat(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5931_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_jc(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5932_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6591440286371151544t_real(aTP_Lamm_jd(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_5933_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5934_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5935_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_5936_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(divide_divide_int(K2),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5937_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu,Nb))) ).

% mask_eq_sum_exp
tff(fact_5938_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu,Nb))) ).

% mask_eq_sum_exp
tff(fact_5939_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5940_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5941_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5942_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_5943_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5944_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5945_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5946_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = uminus_uminus_rat(one_one_rat) ).

% power_minus1_odd
tff(fact_5947_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5948_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_je(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_5949_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_jf(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_5950_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jg(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_5951_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups6591440286371151544t_real(aTP_Lamm_jh(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_5952_int__bit__induct,axiom,
    ! [P: fun_int_o,K2: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K: int] :
              ( aa_int_o(P,K)
             => ( ( K != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K: int] :
                ( aa_int_o(P,K)
               => ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K2) ) ) ) ) ).

% int_bit_induct
tff(fact_5953_eq__diff__eq_H,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z) ) ) ).

% eq_diff_eq'
tff(fact_5954_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_5955_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_5956_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_gt(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5957_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K2)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5958_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_gu(L)),Qr) ).

% divmod_step_int_def
tff(fact_5959_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_5960_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_5961_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_5962_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,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,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_5963_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jj),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5964_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_jk),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5965_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jl),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_5966_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),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_5967_sum__gp,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_complex,
        $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_5968_sum__gp,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_rat,
        $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),minus_minus_rat(one_one_rat,Xb))) ) ).

% sum_gp
tff(fact_5969_sum__gp,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_real,
        $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_real_real(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),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ).

% sum_gp
tff(fact_5970_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( 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,A4))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A4) ) ) ).

% even_set_encode_iff
tff(fact_5971_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_5972_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5973_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5974_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5975_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5976_semiring__norm_I90_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit1,Mb) = aa_num_num(bit1,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(90)
tff(fact_5977_case__prodI,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B))
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B))) ) ).

% case_prodI
tff(fact_5978_case__prodI,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(F,A),B)
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B)) ) ).

% case_prodI
tff(fact_5979_case__prodI,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(F,A),B)
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B)) ) ).

% case_prodI
tff(fact_5980_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) ) ).

% case_prodI
tff(fact_5981_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_5982_case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,C: fun_Co480733329897759773un_o_o] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B3)) )
         => aa_o_o(aa_Cod6379977994291289310un_o_o(C,A3),(B3)) )
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2) ) ).

% case_prodI2
tff(fact_5983_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_num_o(aa_num_fun_num_o(C,A3),B3) )
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).

% case_prodI2
tff(fact_5984_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_num_o(aa_nat_fun_num_o(C,A3),B3) )
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).

% case_prodI2
tff(fact_5985_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A3),B3) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_5986_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A3: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(C,A3),B3) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_5987_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,A: code_integer,B: $o] :
      ( aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A),(B)))
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_5988_mem__case__prodI,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,A: code_integer,B: $o] :
      ( member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A),(B)))
     => member_int(Z,produc1253318751659547953et_int(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_5989_mem__case__prodI,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,A: code_integer,B: $o] :
      ( member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A),(B)))
     => member_real(Z,produc242741666403216561t_real(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_5990_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,A: code_integer,B: $o] :
      ( member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A),(B)))
     => member_complex(Z,produc1043322548047392435omplex(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_5991_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,A: num,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_5992_mem__case__prodI,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,A: num,B: num] :
      ( member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,A),B))
     => member_int(Z,produc6406642877701697732et_int(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_5993_mem__case__prodI,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,A: num,B: num] :
      ( member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,A),B))
     => member_real(Z,produc8296048397933160132t_real(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_5994_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,A: num,B: num] :
      ( member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,A),B))
     => member_complex(Z,produc2866383454006189126omplex(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_5995_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,A: nat,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_5996_mem__case__prodI,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,A: nat,B: num] :
      ( member_int(Z,aa_num_set_int(aa_nat6493022401476965383et_int(C,A),B))
     => member_int(Z,produc9175805072616146554et_int(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_5997_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: nat,C: fun_Co7401227239307276909et_nat] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B3)) )
         => aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A3),(B3))) )
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5998_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: int,C: fun_Co2587678820519358153et_int] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B3)) )
         => member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A3),(B3))) )
     => member_int(Z,produc1253318751659547953et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5999_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: real,C: fun_Co5368275265270938953t_real] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B3)) )
         => member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A3),(B3))) )
     => member_real(Z,produc242741666403216561t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6000_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: complex,C: fun_Co5015990272074005067omplex] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B3)) )
         => member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A3),(B3))) )
     => member_complex(Z,produc1043322548047392435omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6001_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: fun_nu530747996811530292et_nat] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6002_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: fun_nu4940571614878387344et_int] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,A3),B3)) )
     => member_int(Z,produc6406642877701697732et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6003_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: fun_nu4698968662094489488t_real] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,A3),B3)) )
     => member_real(Z,produc8296048397933160132t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6004_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: complex,C: fun_nu9041852840018610706omplex] :
      ( ! [A3: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A3,B3) )
         => member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,A3),B3)) )
     => member_complex(Z,produc2866383454006189126omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6005_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: nat,C: fun_na3896347130304379370et_nat] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6006_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: int,C: fun_na8306170748371236422et_int] :
      ( ! [A3: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A3,B3) )
         => member_int(Z,aa_num_set_int(aa_nat6493022401476965383et_int(C,A3),B3)) )
     => member_int(Z,produc9175805072616146554et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6007_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,Xb: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A3),B3),Xb) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Xb) ) ).

% case_prodI2'
tff(fact_6008_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit0,Mb) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_6009_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit1,Mb) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_6010_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_6011_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa_num_num(bit1,Mb) != one ).

% semiring_norm(86)
tff(fact_6012_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_6013_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_6014_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_6015_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_6016_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_6017_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6018_semiring__norm_I80_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(80)
tff(fact_6019_semiring__norm_I73_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(73)
tff(fact_6020_set__decode__inverse,axiom,
    ! [Nb: nat] : aa_set_nat_nat(nat_set_encode,aa_nat_set_nat(nat_set_decode,Nb)) = Nb ).

% set_decode_inverse
tff(fact_6021_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_6022_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_6023_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_6024_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_6025_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_6026_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6027_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6028_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6029_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6030_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6031_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6032_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6033_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6034_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6035_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6036_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6037_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6038_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6039_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_6040_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6041_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6042_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6043_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_6044_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_6045_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_6046_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = numeral_numeral_real(Nb) ).

% of_nat_numeral
tff(fact_6047_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_6048_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = numeral_numeral_rat(Nb) ).

% of_nat_numeral
tff(fact_6049_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_6050_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_6051_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_6052_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_6053_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_6054_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_6055_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_6056_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_6057_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_6058_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6059_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6060_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6061_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6062_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6063_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6064_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6065_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6066_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6067_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6068_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6069_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6070_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6071_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6072_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xb) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6073_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xb) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6074_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6075_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6076_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Xb) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6077_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6078_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6079_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6080_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6081_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) = aa_nat_rat(semiri681578069525770553at_rat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6082_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),Nb) ).

% of_nat_power
tff(fact_6083_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb) ).

% of_nat_power
tff(fact_6084_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),Nb) ).

% of_nat_power
tff(fact_6085_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb) ).

% of_nat_power
tff(fact_6086_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),Nb) ).

% of_nat_power
tff(fact_6087_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_6088_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_6089_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(9)
tff(fact_6090_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(7)
tff(fact_6091_semiring__norm_I15_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),Nb)) ).

% semiring_norm(15)
tff(fact_6092_semiring__norm_I14_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_6093_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_6094_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_6095_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).

% semiring_norm(70)
tff(fact_6096_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_6097_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_6098_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_rat(semiri681578069525770553at_rat,zero_n2687167440665602831ol_nat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% of_nat_of_bool
tff(fact_6099_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_6100_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_6101_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_6102_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_6103_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_6104_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6105_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6106_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6107_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6108_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).

% of_nat_Suc
tff(fact_6109_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% of_nat_Suc
tff(fact_6110_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_6111_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).

% of_nat_Suc
tff(fact_6112_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_6113_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_6114_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),numeral_numeral_real(W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_6115_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_6116_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_6117_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_6118_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% semiring_norm(4)
tff(fact_6119_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).

% semiring_norm(5)
tff(fact_6120_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one)) ).

% semiring_norm(8)
tff(fact_6121_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)),one)) ).

% semiring_norm(10)
tff(fact_6122_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_6123_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_6124_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_6125_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6126_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6127_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6128_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6129_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6130_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6131_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6132_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6133_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6134_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6135_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6136_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6137_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6138_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6139_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6140_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6141_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6142_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6143_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6144_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6145_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6146_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6147_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6148_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6149_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6150_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6151_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6152_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6153_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6154_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6155_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6156_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6157_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6158_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_6159_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Mb)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_6160_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6161_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6162_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6163_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6164_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_6165_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_6166_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_6167_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_6168_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_6169_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_6170_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_6171_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_6172_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6173_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6174_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6175_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6176_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6177_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6178_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6179_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6180_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6181_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6182_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6183_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6184_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6185_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6186_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6187_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6188_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6189_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6190_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6191_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_6192_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6193_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6194_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6195_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6196_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6197_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6198_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jm),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6199_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_jn),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6200_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_6201_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_6202_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_6203_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_6204_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,P2: produc6271795597528267376eger_o] :
      ( aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2))
     => ~ ! [X4: code_integer,Y3: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
           => ~ aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,X4),(Y3))) ) ) ).

% mem_case_prodE
tff(fact_6205_mem__case__prodE,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,P2: produc6271795597528267376eger_o] :
      ( member_int(Z,produc1253318751659547953et_int(C,P2))
     => ~ ! [X4: code_integer,Y3: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
           => ~ member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,X4),(Y3))) ) ) ).

% mem_case_prodE
tff(fact_6206_mem__case__prodE,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,P2: produc6271795597528267376eger_o] :
      ( member_real(Z,produc242741666403216561t_real(C,P2))
     => ~ ! [X4: code_integer,Y3: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
           => ~ member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,X4),(Y3))) ) ) ).

% mem_case_prodE
tff(fact_6207_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,P2: produc6271795597528267376eger_o] :
      ( member_complex(Z,produc1043322548047392435omplex(C,P2))
     => ~ ! [X4: code_integer,Y3: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
           => ~ member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,X4),(Y3))) ) ) ).

% mem_case_prodE
tff(fact_6208_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
      ( aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2))
     => ~ ! [X4: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X4,Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6209_mem__case__prodE,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
      ( member_int(Z,produc6406642877701697732et_int(C,P2))
     => ~ ! [X4: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X4,Y3) )
           => ~ member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6210_mem__case__prodE,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
      ( member_real(Z,produc8296048397933160132t_real(C,P2))
     => ~ ! [X4: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X4,Y3) )
           => ~ member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6211_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,P2: product_prod_num_num] :
      ( member_complex(Z,produc2866383454006189126omplex(C,P2))
     => ~ ! [X4: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X4,Y3) )
           => ~ member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6212_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,P2: product_prod_nat_num] :
      ( aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P2))
     => ~ ! [X4: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X4,Y3) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6213_mem__case__prodE,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,P2: product_prod_nat_num] :
      ( member_int(Z,produc9175805072616146554et_int(C,P2))
     => ~ ! [X4: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X4,Y3) )
           => ~ member_int(Z,aa_num_set_int(aa_nat6493022401476965383et_int(C,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_6214_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_6215_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_6216_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_6217_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_6218_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_6219_case__prodD,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B)))
     => aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B)) ) ).

% case_prodD
tff(fact_6220_case__prodD,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B))
     => aa_num_o(aa_num_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6221_case__prodD,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B))
     => aa_num_o(aa_nat_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6222_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_6223_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_6224_case__prodE,axiom,
    ! [C: fun_Co480733329897759773un_o_o,P2: produc6271795597528267376eger_o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2)
     => ~ ! [X4: code_integer,Y3: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X4,(Y3)) )
           => ~ aa_o_o(aa_Cod6379977994291289310un_o_o(C,X4),(Y3)) ) ) ).

% case_prodE
tff(fact_6225_case__prodE,axiom,
    ! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
     => ~ ! [X4: num,Y3: num] :
            ( ( P2 = product_Pair_num_num(X4,Y3) )
           => ~ aa_num_o(aa_num_fun_num_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_6226_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
     => ~ ! [X4: nat,Y3: num] :
            ( ( P2 = product_Pair_nat_num(X4,Y3) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_6227_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_6228_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X4: int,Y3: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X4),Y3) ) ) ).

% case_prodE
tff(fact_6229_case__prodD_H,axiom,
    ! [R: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R,A),B),C) ) ).

% case_prodD'
tff(fact_6230_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Z)
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X4),Y3),Z) ) ) ).

% case_prodE'
tff(fact_6231_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_6232_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_6233_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_6234_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_6235_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_6236_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_6237_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_6238_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_6239_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_6240_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_6241_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_6242_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_6243_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_6244_div__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_6245_div__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_6246_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6247_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6248_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6249_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6250_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6251_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6252_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6253_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6254_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_6255_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_6256_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_6257_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_6258_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),one)
           => ( ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N))
             => ( ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N))
               => ( ! [M: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),one)
                 => ( ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N))
                   => ~ ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_6259_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] : Y != aa_num_num(bit0,X22)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_6260_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_6261_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_6262_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_6263_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6264_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6265_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6266_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_6267_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_6268_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_6269_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_6270_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_6271_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_6272_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_6273_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_nat(nat_set_encode,A4) = aa_set_nat_nat(nat_set_encode,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_6274_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_6275_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_6276_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_6277_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_6278_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_6279_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_6280_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,semiri4216267220026989637d_enat(Xb)),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_6281_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_6282_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_6283_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_6284_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = ord_max_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,Y)) ).

% of_nat_max
tff(fact_6285_nat__less__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_6286_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_6287_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_6288_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_real(aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Nb)),numeral_numeral_real(Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_6289_numeral__Bit1,axiom,
    ! [Nb: num] : numeral_numeral_rat(aa_num_num(bit1,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(Nb)),numeral_numeral_rat(Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_6290_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_6291_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_6292_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_6293_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_6294_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_6295_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_6296_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_6297_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6298_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6299_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6300_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6301_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6302_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q3: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6303_power__minus__Bit1,axiom,
    ! [Xb: real,K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_6304_power__minus__Bit1,axiom,
    ! [Xb: int,K2: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_6305_power__minus__Bit1,axiom,
    ! [Xb: complex,K2: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_6306_power__minus__Bit1,axiom,
    ! [Xb: rat,K2: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = uminus_uminus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_6307_power__minus__Bit1,axiom,
    ! [Xb: code_integer,K2: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_6308_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K2 = 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_6309_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_6310_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_6311_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_6312_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_6313_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_6314_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_6315_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_6316_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_6317_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K2),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_6318_real__of__nat__div,axiom,
    ! [D: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_6319_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_6320_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_real(aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= numeral_numeral_real(Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_6321_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numeral_numeral_rat(aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= numeral_numeral_rat(Nb),
        aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_6322_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_6323_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_6324_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6325_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6326_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6327_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6328_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6329_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(nat_set_encode,A4) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_6330_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_6331_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_6332_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_6333_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_6334_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6335_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6336_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6337_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_6338_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_6339_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).

% power3_eq_cube
tff(fact_6340_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),A) ).

% power3_eq_cube
tff(fact_6341_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),A) ).

% power3_eq_cube
tff(fact_6342_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_6343_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_6344_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_6345_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_6346_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6347_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6348_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6349_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K2 = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_6350_pos__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ~ ! [N: nat] :
            ( ( K2 = 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_6351_neg__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K2 = 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_6352_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_6353_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_6354_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K2: 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),K2)
       => 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,K2)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_6355_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)),Mb)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_6356_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_6357_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_6358_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_6359_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_6360_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_6361_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_6362_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_6363_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_6364_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6365_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6366_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6367_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6368_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K2)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_abs_add_iff
tff(fact_6369_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_add_abs_iff
tff(fact_6370_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)
       => ( ! [M: nat] :
              ( 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(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_6371_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_6372_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_6373_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_6374_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_6375_real__of__nat__div3,axiom,
    ! [Nb: 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,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_6376_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_6377_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_6378_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_6379_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_6380_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
                & ( aa_nat_int(F,I2) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6381_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_6382_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_6383_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_6384_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_6385_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_6386_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_6387_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_6388_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6389_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_6390_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum
tff(fact_6391_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum
tff(fact_6392_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum
tff(fact_6393_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum
tff(fact_6394_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_6395_double__arith__series,axiom,
    ! [A: complex,D: complex,Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_jo(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),D))) ).

% double_arith_series
tff(fact_6396_double__arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jp(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D))) ).

% double_arith_series
tff(fact_6397_double__arith__series,axiom,
    ! [A: rat,D: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jq(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),D))) ).

% double_arith_series
tff(fact_6398_double__arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jr(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D))) ).

% double_arith_series
tff(fact_6399_double__arith__series,axiom,
    ! [A: real,D: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_js(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).

% double_arith_series
tff(fact_6400_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_6401_gauss__sum,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6402_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6403_arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jp(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6404_arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jr(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6405_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6406_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6407_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6408_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6409_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6410_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_6411_sum__gp__offset,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_offset
tff(fact_6412_sum__gp__offset,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Mb)),minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),minus_minus_rat(one_one_rat,Xb))) ).

% sum_gp_offset
tff(fact_6413_sum__gp__offset,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),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),Mb)),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,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_6414_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6415_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Mb)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6416_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6417_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_6418_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6419_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6420_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,aa_Pro82899191565270131omplex(produc1917071388513777916omplex(aTP_Lamm_jt),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6421_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_ju),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6422_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_jv),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6423_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_jw,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6424_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_jx),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_6425_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [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)))),E) ) ).

% nat_approx_posE
tff(fact_6426_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_6427_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_6428_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_6429_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_jy,Xb)) ) ).

% monoseq_arctan_series
tff(fact_6430_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(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,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_6431_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(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,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_6432_split__part,axiom,
    ! [P: $o,Q: fun_int_fun_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_jz((P)),Q)),X)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X) ) ) ).

% split_part
tff(fact_6433_Collect__case__prod__mono,axiom,
    ! [A4: fun_int_fun_int_o,B4: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A4,B4)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A4)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B4))) ) ).

% Collect_case_prod_mono
tff(fact_6434_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_ka),Prod) ).

% prod.disc_eq_case
tff(fact_6435_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_6436_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_6437_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_6438_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_6439_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_6440_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_6441_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_1: nat] : aa_nat_o(P,X_1)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_6442_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_6443_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(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_6444_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_6445_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_6446_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_6447_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_6448_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_6449_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_6450_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_6451_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_6452_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_6453_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_6454_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_6455_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_6456_norm__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% norm_minus_commute
tff(fact_6457_norm__minus__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),A)) ).

% norm_minus_commute
tff(fact_6458_norm__mult,axiom,
    ! [Xb: real,Y: 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
tff(fact_6459_norm__mult,axiom,
    ! [Xb: complex,Y: complex] : 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
tff(fact_6460_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_6461_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_6462_norm__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb) ).

% norm_power
tff(fact_6463_norm__power,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb) ).

% norm_power
tff(fact_6464_norm__uminus__minus,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).

% norm_uminus_minus
tff(fact_6465_norm__uminus__minus,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).

% norm_uminus_minus
tff(fact_6466_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6467_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_6468_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6469_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6470_norm__mult__less,axiom,
    ! [Xb: real,R2: real,Y: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2)
       => 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),S2)) ) ) ).

% norm_mult_less
tff(fact_6471_norm__mult__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2)
       => 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),S2)) ) ) ).

% norm_mult_less
tff(fact_6472_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_6473_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_6474_norm__triangle__lt,axiom,
    ! [Xb: real,Y: real,E: 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))),E)
     => 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))),E) ) ).

% norm_triangle_lt
tff(fact_6475_norm__triangle__lt,axiom,
    ! [Xb: complex,Y: complex,E: 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))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),E) ) ).

% norm_triangle_lt
tff(fact_6476_norm__add__less,axiom,
    ! [Xb: real,R2: real,Y: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2)
       => 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),S2)) ) ) ).

% norm_add_less
tff(fact_6477_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2)
       => 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),S2)) ) ) ).

% norm_add_less
tff(fact_6478_norm__power__ineq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_6479_norm__power__ineq,axiom,
    ! [Xb: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_6480_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).

% norm_triangle_mono
tff(fact_6481_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).

% norm_triangle_mono
tff(fact_6482_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_6483_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_6484_norm__triangle__le,axiom,
    ! [Xb: real,Y: real,E: 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))),E)
     => 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))),E) ) ).

% norm_triangle_le
tff(fact_6485_norm__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E: 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))),E)
     => 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))),E) ) ).

% norm_triangle_le
tff(fact_6486_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_6487_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_6488_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_6489_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_6490_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_6491_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_6492_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))) ).

% norm_triangle_ineq4
tff(fact_6493_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))) ).

% norm_triangle_ineq4
tff(fact_6494_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_6495_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_6496_norm__triangle__le__diff,axiom,
    ! [Xb: real,Y: real,E: 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))),E)
     => 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))),E) ) ).

% norm_triangle_le_diff
tff(fact_6497_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Y: complex,E: 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))),E)
     => 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))),E) ) ).

% norm_triangle_le_diff
tff(fact_6498_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ).

% norm_diff_ineq
tff(fact_6499_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))) ).

% norm_diff_ineq
tff(fact_6500_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% norm_triangle_ineq2
tff(fact_6501_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))) ).

% norm_triangle_ineq2
tff(fact_6502_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6503_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6504_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6505_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6506_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% norm_triangle_ineq3
tff(fact_6507_norm__triangle__ineq3,axiom,
    ! [A: complex,B: 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,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))) ).

% norm_triangle_ineq3
tff(fact_6508_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_6509_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_6510_norm__power__diff,axiom,
    ! [Z: real,W: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_6511_norm__power__diff,axiom,
    ! [Z: complex,W: complex,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),Mb)),aa_nat_complex(power_power_complex(W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_6512_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_6513_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),numeral_numeral_real(aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_kb,Xb)) ) ) ) ).

% ln_series
tff(fact_6514_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6515_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_6516_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_6517_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6518_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6519_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_kc,Xb)) ) ) ).

% arctan_series
tff(fact_6520_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ke(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6521_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Nb: nat] :
      ( ( H != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),H)),Nb),aa_nat_rat(power_power_rat(Z),Nb)),H),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kg(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6522_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: 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)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ki(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6523_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_kc,Xb)) ) ).

% summable_arctan_series
tff(fact_6524_lessThan__iff,axiom,
    ! [I: rat,K2: rat] :
      ( member_rat(I,set_ord_lessThan_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K2) ) ).

% lessThan_iff
tff(fact_6525_lessThan__iff,axiom,
    ! [I: num,K2: num] :
      ( member_num(I,set_ord_lessThan_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I),K2) ) ).

% lessThan_iff
tff(fact_6526_lessThan__iff,axiom,
    ! [I: int,K2: int] :
      ( member_int(I,set_ord_lessThan_int(K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K2) ) ).

% lessThan_iff
tff(fact_6527_lessThan__iff,axiom,
    ! [I: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2) ) ).

% lessThan_iff
tff(fact_6528_lessThan__iff,axiom,
    ! [I: real,K2: real] :
      ( member_real(I,set_or5984915006950818249n_real(K2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K2) ) ).

% lessThan_iff
tff(fact_6529_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_6530_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_6531_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_6532_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_6533_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_6534_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_6535_lessThan__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( 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_6536_lessThan__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_6537_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_6538_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_6539_eq__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K2) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_6540_Suc__eq__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K2) )
    <=> ( Nb = aa_num_nat(pred_numeral,K2) ) ) ).

% Suc_eq_numeral
tff(fact_6541_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6542_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6543_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6544_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_6545_less__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% less_numeral_Suc
tff(fact_6546_less__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% less_Suc_numeral
tff(fact_6547_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2)) ).

% pred_numeral_simps(3)
tff(fact_6548_le__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% le_Suc_numeral
tff(fact_6549_le__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% le_numeral_Suc
tff(fact_6550_diff__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K2)),Nb) ).

% diff_numeral_Suc
tff(fact_6551_diff__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K2)) ).

% diff_Suc_numeral
tff(fact_6552_max__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K2)),Nb)) ).

% max_numeral_Suc
tff(fact_6553_max__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K2))) ).

% max_Suc_numeral
tff(fact_6554_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_6555_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_6556_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_6557_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_6558_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6559_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_gy(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_6560_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_ha(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_6561_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_6562_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_6563_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_6564_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_kj(U)) ).

% lessThan_def
tff(fact_6565_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_kk,U)) ).

% lessThan_def
tff(fact_6566_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_kl,U)) ).

% lessThan_def
tff(fact_6567_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_km,U)) ).

% lessThan_def
tff(fact_6568_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu,U)) ).

% lessThan_def
tff(fact_6569_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_kn,U)) ).

% lessThan_def
tff(fact_6570_powser__insidea,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(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_kp(F),Z)) ) ) ).

% powser_insidea
tff(fact_6571_powser__insidea,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(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_kr(F),Z)) ) ) ).

% powser_insidea
tff(fact_6572_lessThan__strict__subset__iff,axiom,
    ! [Mb: rat,Nb: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(Mb),set_ord_lessThan_rat(Nb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6573_lessThan__strict__subset__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_less_set_num(set_ord_lessThan_num(Mb),set_ord_lessThan_num(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6574_lessThan__strict__subset__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ord_less_set_int(set_ord_lessThan_int(Mb),set_ord_lessThan_int(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6575_lessThan__strict__subset__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6576_lessThan__strict__subset__iff,axiom,
    ! [Mb: real,Nb: real] :
      ( ord_less_set_real(set_or5984915006950818249n_real(Mb),set_or5984915006950818249n_real(Nb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_6577_numeral__eq__Suc,axiom,
    ! [K2: num] : aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K2)) ).

% numeral_eq_Suc
tff(fact_6578_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( ! [D2: 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,K2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),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))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_6579_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_6580_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_kt(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_6581_pred__numeral__def,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),one_one_nat) ).

% pred_numeral_def
tff(fact_6582_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Nb: real] :
      ( ! [X4: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X4)),aa_real_nat(P,X4))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Nb))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Nb))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_hu(Q),P),set_or5984915006950818249n_real(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_6583_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Nb: nat] :
      ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P,X4))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hx(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_6584_powser__inside,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(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_ko(F),Z)) ) ) ).

% powser_inside
tff(fact_6585_powser__inside,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(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_kq(F),Z)) ) ) ).

% powser_inside
tff(fact_6586_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6587_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6588_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hy(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6589_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_6590_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,Nb: nat,R2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ku(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_6591_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,Nb: nat,R2: rat] : minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),R2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_kv(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_6592_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,Nb: nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),R2)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_kw(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_6593_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_kx(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6594_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_ky(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6595_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_kz(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_6596_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ik(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = minus_minus_rat(aa_nat_rat(F,Mb),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6597_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_il(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6598_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : groups6591440286371151544t_real(aTP_Lamm_im(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_6599_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hy(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_6600_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups6591440286371151544t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_6601_power__diff__1__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),one_one_complex)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6602_power__diff__1__eq,axiom,
    ! [Xb: rat,Nb: nat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xb,one_one_rat)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6603_power__diff__1__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6604_power__diff__1__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),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)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_6605_one__diff__power__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6606_one__diff__power__eq,axiom,
    ! [Xb: rat,Nb: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6607_one__diff__power__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6608_one__diff__power__eq,axiom,
    ! [Xb: real,Nb: 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),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_6609_geometric__sum,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( Xb != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_6610_geometric__sum,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb),one_one_rat),minus_minus_rat(Xb,one_one_rat)) ) ) ).

% geometric_sum
tff(fact_6611_geometric__sum,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != one_one_real )
     => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = 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),Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ) ).

% geometric_sum
tff(fact_6612_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_6613_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_6614_dbl__dec__def,axiom,
    ! [Xb: rat] : neg_nu3179335615603231917ec_rat(Xb) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Xb),one_one_rat) ).

% dbl_dec_def
tff(fact_6615_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_6616_sum__gp__strict,axiom,
    ! [Xb: complex,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_strict
tff(fact_6617_sum__gp__strict,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),Nb)),minus_minus_rat(one_one_rat,Xb))) ).

% sum_gp_strict
tff(fact_6618_sum__gp__strict,axiom,
    ! [Xb: real,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),aa_real_real(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),Nb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_strict
tff(fact_6619_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_la(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lb(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6620_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lc(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ld(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6621_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_le(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lf(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6622_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_lg(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_lh(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_6623_power__diff__sumr2,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_li(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6624_power__diff__sumr2,axiom,
    ! [Xb: rat,Nb: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb),aa_nat_rat(power_power_rat(Y),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xb,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_lj(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6625_power__diff__sumr2,axiom,
    ! [Xb: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) = 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_lk(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6626_power__diff__sumr2,axiom,
    ! [Xb: real,Nb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ll(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_6627_diff__power__eq__sum,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_lm(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6628_diff__power__eq__sum,axiom,
    ! [Xb: rat,Nb: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xb,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ln(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6629_diff__power__eq__sum,axiom,
    ! [Xb: int,Nb: 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,Nb))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_lo(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6630_diff__power__eq__sum,axiom,
    ! [Xb: real,Nb: 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,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lp(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_6631_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_rat,K5: rat,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),K5) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6632_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K5: int,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),K5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6633_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K5: nat,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6634_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K5: real,K2: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),K5) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K5)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_6635_one__diff__power__eq_H,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_lq(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6636_one__diff__power__eq_H,axiom,
    ! [Xb: rat,Nb: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xb)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_lr(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6637_one__diff__power__eq_H,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ls(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6638_one__diff__power__eq_H,axiom,
    ! [Xb: real,Nb: 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),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lt(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_6639_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lu(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_lv(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),groups6591440286371151544t_real(aTP_Lamm_lw(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_6640_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_6641_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_6642_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_lx(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_6643_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_gf(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_6644_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_6645_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_6646_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ly(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_6647_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fw(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_6648_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Nb: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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,Nb))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_6649_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Nb: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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,Nb))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_6650_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Nb: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_6651_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_6652_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_6653_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_6654_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_lz(F),C)) ) ).

% summable_mult
tff(fact_6655_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fs(F),C)) ) ).

% summable_mult2
tff(fact_6656_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(F),G)) ) ) ).

% summable_add
tff(fact_6657_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fz(F),G)) ) ) ).

% summable_add
tff(fact_6658_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ma(F),G)) ) ) ).

% summable_add
tff(fact_6659_summable__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_gd(F),G)) ) ) ).

% summable_diff
tff(fact_6660_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_lx(F),C)) ) ).

% summable_divide
tff(fact_6661_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_gf(F),C)) ) ).

% summable_divide
tff(fact_6662_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_hz(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_6663_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2)) ) ).

% summable_ignore_initial_segment
tff(fact_6664_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_6665_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_6666_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_6667_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ly(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_6668_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fw(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_6669_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_6670_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_6671_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_6672_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_lz(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_6673_suminf__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_fs(F),C)) ) ) ).

% suminf_mult2
tff(fact_6674_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(F),G)) ) ) ) ).

% suminf_add
tff(fact_6675_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fz(F),G)) ) ) ) ).

% suminf_add
tff(fact_6676_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_ma(F),G)) ) ) ) ).

% suminf_add
tff(fact_6677_suminf__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_gd(F),G)) ) ) ) ).

% suminf_diff
tff(fact_6678_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_lx(F),C)) = aa_complex_complex(divide1717551699836669952omplex(suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_6679_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_gf(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_6680_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_6681_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_6682_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_6683_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_6684_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_6685_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_6686_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_6687_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_6688_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_6689_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gy(F)) ).

% summable_zero_power'
tff(fact_6690_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_ha(F)) ).

% summable_zero_power'
tff(fact_6691_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_mb(F)) ).

% summable_zero_power'
tff(fact_6692_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gy(F)) ).

% summable_0_powser
tff(fact_6693_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_ha(F)) ).

% summable_0_powser
tff(fact_6694_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_mc(F),Z)) ) ).

% powser_split_head(3)
tff(fact_6695_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_md(F),Z)) ) ).

% powser_split_head(3)
tff(fact_6696_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_mc(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(F),Z)) ) ).

% summable_powser_split_head
tff(fact_6697_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_md(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(F),Z)) ) ).

% summable_powser_split_head
tff(fact_6698_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_me(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_6699_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,Mb: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mf(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_6700_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_mg(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_6701_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_gx(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_6702_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_6703_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_6704_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_6705_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_6706_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_6707_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_6708_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_6709_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_6710_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,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_6711_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_6712_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_6713_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_6714_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_6715_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_6716_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_6717_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,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_6718_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_6719_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_6720_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_6721_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_6722_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_hz(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_6723_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I6: set_nat] :
      ( summable_int(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => 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,I6)),suminf_int(F)) ) ) ) ).

% sum_le_suminf
tff(fact_6724_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I6: set_nat] :
      ( summable_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => 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),I6)),suminf_nat(F)) ) ) ) ).

% sum_le_suminf
tff(fact_6725_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I6: set_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => 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,groups6591440286371151544t_real(F,I6)),suminf_real(F)) ) ) ) ).

% sum_le_suminf
tff(fact_6726_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ) ).

% suminf_split_initial_segment
tff(fact_6727_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),K2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K2))) ) ) ).

% suminf_minus_initial_segment
tff(fact_6728_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Nb: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_6729_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_6730_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Nb: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_6731_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(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_mc(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_6732_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ko(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_md(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_6733_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_mc(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_6734_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_md(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ko(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_6735_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E: real] :
      ( summable_complex(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N7)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_6736_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E: real] :
      ( summable_real(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N7)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_6737_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_real(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_6738_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_complex(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mh(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_6739_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_ko(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_6740_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R2),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M6)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_mi(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_6741_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hy(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6742_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_hz(F),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_6743_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_mj,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_6744_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_mk) ).

% pi_series
tff(fact_6745_pred__subset__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ord_le2909477399203599485un_o_o(aTP_Lamm_ml(R),aTP_Lamm_ml(S3))
    <=> ord_le8980329558974975238eger_o(R,S3) ) ).

% pred_subset_eq2
tff(fact_6746_pred__subset__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ord_le3306795289226699318_num_o(aTP_Lamm_mm(R),aTP_Lamm_mm(S3))
    <=> ord_le880128212290418581um_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_6747_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aTP_Lamm_mn(R),aTP_Lamm_mn(S3))
    <=> ord_le8085105155179020875at_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_6748_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_mo(R),aTP_Lamm_mo(S3))
    <=> ord_le3146513528884898305at_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_6749_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_mp(R),aTP_Lamm_mp(S3))
    <=> ord_le2843351958646193337nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_6750_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_6751_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = numeral_numeral_real(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6752_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6753_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_6754_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_mq,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_6755_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => aa_complex_o(sums_complex(aTP_Lamm_mr(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_6756_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_6757_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_6758_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_6759_dbl__inc__simps_I2_J,axiom,
    neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_6760_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_6761_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_6762_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_6763_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_6764_dbl__inc__simps_I4_J,axiom,
    neg_nu5219082963157363817nc_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_6765_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_6766_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6767_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K2)) = numeral_numeral_real(aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6768_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K2)) = numeral_numeral_rat(aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6769_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_6770_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K2))) ).

% dbl_dec_simps(1)
tff(fact_6771_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_dec_simps(1)
tff(fact_6772_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2))) ).

% dbl_dec_simps(1)
tff(fact_6773_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : neg_nu3179335615603231917ec_rat(uminus_uminus_rat(numeral_numeral_rat(K2))) = uminus_uminus_rat(neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K2))) ).

% dbl_dec_simps(1)
tff(fact_6774_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K2))) ).

% dbl_dec_simps(1)
tff(fact_6775_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K2))) ).

% dbl_inc_simps(1)
tff(fact_6776_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_inc_simps(1)
tff(fact_6777_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K2))) ).

% dbl_inc_simps(1)
tff(fact_6778_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : neg_nu5219082963157363817nc_rat(uminus_uminus_rat(numeral_numeral_rat(K2))) = uminus_uminus_rat(neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K2))) ).

% dbl_inc_simps(1)
tff(fact_6779_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K2))) ).

% dbl_inc_simps(1)
tff(fact_6780_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_gy(A)),Xb)
    <=> ( aa_nat_complex(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_6781_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_ha(A)),Xb)
    <=> ( aa_nat_real(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_6782_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S2: real,Ta: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( aa_real_o(sums_real(F),S2)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S2),Ta) ) ) ) ).

% sums_le
tff(fact_6783_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S2: nat,Ta: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( aa_nat_o(sums_nat(F),S2)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S2),Ta) ) ) ) ).

% sums_le
tff(fact_6784_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S2: int,Ta: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( aa_int_o(sums_int(F),S2)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S2),Ta) ) ) ) ).

% sums_le
tff(fact_6785_sums__mult2,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_fs(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ).

% sums_mult2
tff(fact_6786_sums__mult,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_lz(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ) ).

% sums_mult
tff(fact_6787_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% sums_add
tff(fact_6788_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( aa_nat_o(sums_nat(F),A)
     => ( aa_nat_o(sums_nat(G),B)
       => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fz(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_6789_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( aa_int_o(sums_int(F),A)
     => ( aa_int_o(sums_int(G),B)
       => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ma(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_6790_sums__diff,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_gd(F),G)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% sums_diff
tff(fact_6791_sums__divide,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_lx(F),C)),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ).

% sums_divide
tff(fact_6792_sums__divide,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_gf(F),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ).

% sums_divide
tff(fact_6793_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ms(C),F)),aa_complex_complex(times_times_complex(D),C))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_6794_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_mt(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_6795_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ly(C),F)),aa_complex_complex(times_times_complex(C),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_6796_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_fw(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_6797_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ly(C),F)),A)
     => ( ( C != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ) ).

% sums_mult_D
tff(fact_6798_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_fw(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% sums_mult_D
tff(fact_6799_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S2: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_complex_o(sums_complex(aTP_Lamm_mu(F)),S2)
       => aa_complex_o(sums_complex(F),S2) ) ) ).

% sums_Suc_imp
tff(fact_6800_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_hz(F)),S2)
       => aa_real_o(sums_real(F),S2) ) ) ).

% sums_Suc_imp
tff(fact_6801_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( aa_real_o(sums_real(aTP_Lamm_hz(F)),S2)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_6802_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_hz(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_6803_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_hy(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_6804_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_ij(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_6805_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_complex,S2: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_complex(F,I2) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mv(Nb),F)),S2)
      <=> aa_complex_o(sums_complex(F),S2) ) ) ).

% sums_zero_iff_shift
tff(fact_6806_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,S2: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_mw(Nb),F)),S2)
      <=> aa_real_o(sums_real(F),S2) ) ) ).

% sums_zero_iff_shift
tff(fact_6807_powser__sums__if,axiom,
    ! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_mx(Mb),Z)),aa_nat_complex(power_power_complex(Z),Mb)) ).

% powser_sums_if
tff(fact_6808_powser__sums__if,axiom,
    ! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_my(Mb),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)) ).

% powser_sums_if
tff(fact_6809_powser__sums__if,axiom,
    ! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_mz(Mb),Z)),aa_nat_int(power_power_int(Z),Mb)) ).

% powser_sums_if
tff(fact_6810_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_gy(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_6811_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_ha(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_6812_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Nb: nat,S2: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),Nb)),S2)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_iff_shift
tff(fact_6813_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Nb: nat,S2: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
    <=> aa_real_o(sums_real(F),S2) ) ).

% sums_iff_shift'
tff(fact_6814_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S2: real,Nb: nat] :
      ( aa_real_o(sums_real(F),S2)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ib(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_split_initial_segment
tff(fact_6815_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S3: real,A4: set_nat,S4: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S3)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( ( S4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(G),F),A4)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_nb(G),A4),F)),S4) ) ) ) ).

% sums_If_finite_set'
tff(fact_6816_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_6817_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_6818_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != numeral_numeral_real(aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_6819_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_6820_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_6821_dbl__inc__def,axiom,
    ! [Xb: rat] : neg_nu5219082963157363817nc_rat(Xb) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Xb)),one_one_rat) ).

% dbl_inc_def
tff(fact_6822_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_6823_pred__equals__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ! [X3: code_integer,Xa2: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Xa2)),R)
        <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Xa2)),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6824_pred__equals__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ! [X3: num,Xa2: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X3,Xa2),R)
        <=> member7279096912039735102um_num(product_Pair_num_num(X3,Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6825_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X3: nat,Xa2: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X3,Xa2),R)
        <=> member9148766508732265716at_num(product_Pair_nat_num(X3,Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6826_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Xa2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2),R)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6827_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Xa2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6828_bot__empty__eq2,axiom,
    ! [X: code_integer,Xa: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(bot_bo897209107778963633un_o_o,X),(Xa))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Xa)),bot_bo5379713665208646970eger_o) ) ).

% bot_empty_eq2
tff(fact_6829_bot__empty__eq2,axiom,
    ! [X: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(bot_bo3657453288633029378_num_o,X),Xa)
    <=> member7279096912039735102um_num(product_Pair_num_num(X,Xa),bot_bo9056780473022590049um_num) ) ).

% bot_empty_eq2
tff(fact_6830_bot__empty__eq2,axiom,
    ! [X: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X),Xa)
    <=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_6831_bot__empty__eq2,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X),Xa)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_6832_bot__empty__eq2,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_6833_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_6834_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_6835_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_6836_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_6837_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_6838_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_6839_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_6840_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_6841_geometric__sums,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C))) ) ).

% geometric_sums
tff(fact_6842_geometric__sums,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => aa_complex_o(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C))) ) ).

% geometric_sums
tff(fact_6843_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_nc),one_one_real) ).

% power_half_series
tff(fact_6844_subrelI,axiom,
    ! [R2: set_Pr448751882837621926eger_o,S2: set_Pr448751882837621926eger_o] :
      ( ! [X4: code_integer,Y3: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X4,(Y3)),R2)
         => member1379723562493234055eger_o(produc6677183202524767010eger_o(X4,(Y3)),S2) )
     => ord_le8980329558974975238eger_o(R2,S2) ) ).

% subrelI
tff(fact_6845_subrelI,axiom,
    ! [R2: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num] :
      ( ! [X4: num,Y3: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X4,Y3),R2)
         => member7279096912039735102um_num(product_Pair_num_num(X4,Y3),S2) )
     => ord_le880128212290418581um_num(R2,S2) ) ).

% subrelI
tff(fact_6846_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ! [X4: nat,Y3: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X4,Y3),R2)
         => member9148766508732265716at_num(product_Pair_nat_num(X4,Y3),S2) )
     => ord_le8085105155179020875at_num(R2,S2) ) ).

% subrelI
tff(fact_6847_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3),S2) )
     => ord_le3146513528884898305at_nat(R2,S2) ) ).

% subrelI
tff(fact_6848_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y3),S2) )
     => ord_le2843351958646193337nt_int(R2,S2) ) ).

% subrelI
tff(fact_6849_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_6850_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_6851_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_6852_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_nd(G)),Xb) ) ).

% sums_if'
tff(fact_6853_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_ne(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sums_if
tff(fact_6854_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_6855_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_6856_sin__cos__npi,axiom,
    ! [Nb: 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))),Nb)))),pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_6857_diffs__equiv,axiom,
    ! [C: fun_nat_complex,Xb: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_nf(C),Xb))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ng(C),Xb)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_nf(C),Xb))) ) ).

% diffs_equiv
tff(fact_6858_diffs__equiv,axiom,
    ! [C: fun_nat_real,Xb: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_nh(C),Xb))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ni(C),Xb)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_nh(C),Xb))) ) ).

% diffs_equiv
tff(fact_6859_cos__pi__eq__zero,axiom,
    ! [Mb: 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))),Mb))))),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_6860_monoI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M)),aa_nat_real(X6,N)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI1
tff(fact_6861_monoI1,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M)),aa_nat_set_nat(X6,N)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI1
tff(fact_6862_monoI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M)),aa_nat_rat(X6,N)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI1
tff(fact_6863_monoI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,M)),aa_nat_num(X6,N)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI1
tff(fact_6864_monoI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M)),aa_nat_nat(X6,N)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI1
tff(fact_6865_monoI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M)),aa_nat_int(X6,N)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI1
tff(fact_6866_monoI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,M)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI2
tff(fact_6867_monoI2,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,M)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI2
tff(fact_6868_monoI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,M)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI2
tff(fact_6869_monoI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N)),aa_nat_num(X6,M)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI2
tff(fact_6870_monoI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,M)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI2
tff(fact_6871_monoI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,M)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI2
tff(fact_6872_monoseq__def,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M5)),aa_nat_real(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N4)),aa_nat_real(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_6873_monoseq__def,axiom,
    ! [X6: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M5)),aa_nat_set_nat(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N4)),aa_nat_set_nat(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_6874_monoseq__def,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M5)),aa_nat_rat(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N4)),aa_nat_rat(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_6875_monoseq__def,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,M5)),aa_nat_num(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N4)),aa_nat_num(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_6876_monoseq__def,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M5)),aa_nat_nat(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N4)),aa_nat_nat(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_6877_monoseq__def,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M5)),aa_nat_int(X6,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N4)),aa_nat_int(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_6878_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_6879_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_6880_sin__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(sin_real,Xb) ).

% sin_pi_minus
tff(fact_6881_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_6882_cos__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_pi_minus
tff(fact_6883_cos__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_minus_pi
tff(fact_6884_sin__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_minus_pi
tff(fact_6885_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_6886_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_6887_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_6888_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_6889_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_6890_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_6891_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_6892_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_6893_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_6894_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_6895_cos__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).

% cos_2pi_minus
tff(fact_6896_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_6897_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_6898_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_6899_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_6900_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_6901_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_6902_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_6903_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_6904_sin__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_2pi_minus
tff(fact_6905_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_6906_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_6907_polar__Ex,axiom,
    ! [Xb: real,Y: real] :
    ? [R4: real,A3: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),aa_real_real(cos_real,A3)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R4),aa_real_real(sin_real,A3)) ) ) ).

% polar_Ex
tff(fact_6908_sin__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_diff
tff(fact_6909_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_6910_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_6911_sin__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = 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(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_6912_cos__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_6913_cos__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = 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,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_6914_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_6915_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_6916_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_6917_sin__double,axiom,
    ! [Xb: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,Xb))),aa_complex_complex(cos_complex,Xb)) ).

% sin_double
tff(fact_6918_sin__double,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,Xb))),aa_real_real(cos_real,Xb)) ).

% sin_double
tff(fact_6919_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_6920_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_6921_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_6922_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_6923_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_6924_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_6925_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_6926_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_6927_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_6928_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_6929_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_6930_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_6931_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_6932_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_6933_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_6934_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_6935_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_6936_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_6937_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_6938_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_6939_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_6940_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_6941_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_6942_cos__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_6943_cos__double,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_6944_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_6945_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_6946_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_6947_diffs__def,axiom,
    ! [C: fun_nat_int,X: nat] : aa_nat_int(diffs_int(C),X) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X))),aa_nat_int(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_6948_diffs__def,axiom,
    ! [C: fun_nat_real,X: nat] : aa_nat_real(diffs_real(C),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X))),aa_nat_real(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_6949_diffs__def,axiom,
    ! [C: fun_nat_rat,X: nat] : aa_nat_rat(diffs_rat(C),X) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X))),aa_nat_rat(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_6950_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 )
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),pi)
            & ( Xb = aa_real_real(cos_real,T3) )
            & ( Y = aa_real_real(sin_real,T3) ) ) ) ) ).

% sincos_total_pi
tff(fact_6951_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: 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,Mb))),pi)),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,Mb)),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_6952_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: 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,Mb))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_6953_termdiff__converges__all,axiom,
    ! [C: fun_nat_complex,Xb: complex] :
      ( ! [X4: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),X4))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_nf(C),Xb)) ) ).

% termdiff_converges_all
tff(fact_6954_termdiff__converges__all,axiom,
    ! [C: fun_nat_real,Xb: real] :
      ( ! [X4: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),X4))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_nh(C),Xb)) ) ).

% termdiff_converges_all
tff(fact_6955_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),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_6956_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_6957_cos__is__zero,axiom,
    ? [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),numeral_numeral_real(aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X4) = 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),numeral_numeral_real(aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
tff(fact_6958_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_6959_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)
       => ? [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),pi)
            & ( aa_real_real(cos_real,X4) = 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 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_6960_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 )
         => ? [T3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T3) )
              & ( Y = aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_6961_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 )
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T3) )
          & ( Y = aa_real_real(sin_real,T3) ) ) ) ).

% sincos_total_2pi_le
tff(fact_6962_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 )
     => ~ ! [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T3) )
               => ( Y != aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_6963_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_6964_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_6965_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_6966_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_6967_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),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_6968_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,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_6969_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),numeral_numeral_real(aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_6970_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_30
tff(fact_6971_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),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_6972_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),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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_6973_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),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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,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_6974_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),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),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_6975_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_60
tff(fact_6976_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_6977_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_6978_cos__treble__cos,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),Xb)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,Xb))) ).

% cos_treble_cos
tff(fact_6979_cos__treble__cos,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),aa_real_real(cos_real,Xb))) ).

% cos_treble_cos
tff(fact_6980_termdiff__converges,axiom,
    ! [Xb: real,K5: real,C: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),K5)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X4)),K5)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),X4)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_nj(Xb),C)) ) ) ).

% termdiff_converges
tff(fact_6981_termdiff__converges,axiom,
    ! [Xb: complex,K5: real,C: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),K5)
     => ( ! [X4: complex] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X4)),K5)
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),X4)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nk(Xb),C)) ) ) ).

% termdiff_converges
tff(fact_6982_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,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_6983_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)),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_6984_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),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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,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_6985_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),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),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_6986_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)
       => ? [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X4) = 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),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),numeral_numeral_real(aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_6987_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),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),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_6988_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),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),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_6989_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ( ? [X3: nat] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)
        | ? [X3: nat] : Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_6990_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_6991_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),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_6992_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_6993_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),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_6994_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_6995_mono__SucI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,aa_nat_nat(suc,N)))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI1
tff(fact_6996_mono__SucI1,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,aa_nat_nat(suc,N)))
     => topolo7278393974255667507et_nat(X6) ) ).

% mono_SucI1
tff(fact_6997_mono__SucI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,aa_nat_nat(suc,N)))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI1
tff(fact_6998_mono__SucI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N)),aa_nat_num(X6,aa_nat_nat(suc,N)))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI1
tff(fact_6999_mono__SucI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,aa_nat_nat(suc,N)))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI1
tff(fact_7000_mono__SucI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,aa_nat_nat(suc,N)))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI1
tff(fact_7001_mono__SucI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N))),aa_nat_real(X6,N))
     => topolo6980174941875973593q_real(X6) ) ).

% mono_SucI2
tff(fact_7002_mono__SucI2,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N))),aa_nat_set_nat(X6,N))
     => topolo7278393974255667507et_nat(X6) ) ).

% mono_SucI2
tff(fact_7003_mono__SucI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N))),aa_nat_rat(X6,N))
     => topolo4267028734544971653eq_rat(X6) ) ).

% mono_SucI2
tff(fact_7004_mono__SucI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N))),aa_nat_num(X6,N))
     => topolo1459490580787246023eq_num(X6) ) ).

% mono_SucI2
tff(fact_7005_mono__SucI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N))),aa_nat_nat(X6,N))
     => topolo4902158794631467389eq_nat(X6) ) ).

% mono_SucI2
tff(fact_7006_mono__SucI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N))),aa_nat_int(X6,N))
     => topolo4899668324122417113eq_int(X6) ) ).

% mono_SucI2
tff(fact_7007_monoseq__Suc,axiom,
    ! [X6: fun_nat_real] :
      ( topolo6980174941875973593q_real(X6)
    <=> ( ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N4)),aa_nat_real(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,aa_nat_nat(suc,N4))),aa_nat_real(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_7008_monoseq__Suc,axiom,
    ! [X6: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X6)
    <=> ( ! [N4: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N4)),aa_nat_set_nat(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,aa_nat_nat(suc,N4))),aa_nat_set_nat(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_7009_monoseq__Suc,axiom,
    ! [X6: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X6)
    <=> ( ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N4)),aa_nat_rat(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,aa_nat_nat(suc,N4))),aa_nat_rat(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_7010_monoseq__Suc,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [N4: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N4)),aa_nat_num(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N4))),aa_nat_num(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_7011_monoseq__Suc,axiom,
    ! [X6: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X6)
    <=> ( ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N4)),aa_nat_nat(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,aa_nat_nat(suc,N4))),aa_nat_nat(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_7012_monoseq__Suc,axiom,
    ! [X6: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X6)
    <=> ( ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N4)),aa_nat_int(X6,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,aa_nat_nat(suc,N4))),aa_nat_int(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_7013_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nl,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_7014_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nl,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_7015_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nl,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_7016_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_nm,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_7017_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_7018_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,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,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,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_7019_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(cos_real,T3)),aa_real_real(sin_real,T3)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_7020_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_7021_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_7022_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_7023_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_7024_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_7025_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_7026_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_7027_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_7028_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_7029_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_7030_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7031_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_7032_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7033_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7034_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7035_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_7036_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_7037_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_7038_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_7039_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_7040_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_n
tff(fact_7041_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_nat
tff(fact_7042_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7043_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7044_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_7045_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7046_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_7047_norm__cos__sin,axiom,
    ! [Ta: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(cos_real,Ta)),aa_real_real(sin_real,Ta))) = one_one_real ).

% norm_cos_sin
tff(fact_7048_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_7049_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(minus_minus_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,C),D)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ).

% complex_diff
tff(fact_7050_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_7051_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_7052_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_7053_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_7054_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_gt_zero
tff(fact_7055_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_7056_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_7057_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_7058_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).

% fact_not_neg
tff(fact_7059_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_7060_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_7061_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_7062_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_7063_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_7064_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_7065_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_7066_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_7067_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_7068_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_7069_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_7070_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_7071_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).

% fact_dvd
tff(fact_7072_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_7073_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_7074_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_7075_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_7076_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_7077_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_7078_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_7079_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_7080_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7081_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7082_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7083_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: 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,K2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7084_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_7085_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7086_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7087_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7088_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_7089_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,C),D)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).

% complex_mult
tff(fact_7090_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_7091_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(aa_rea2649881322057434177omplex(complex2,one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_7092_tan__def,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X)),aa_complex_complex(cos_complex,X)) ).

% tan_def
tff(fact_7093_tan__def,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,X)) ).

% tan_def
tff(fact_7094_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_7095_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_7096_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_7097_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => 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,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_7098_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_7099_fact__numeral,axiom,
    ! [K2: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_7100_fact__numeral,axiom,
    ! [K2: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_7101_fact__numeral,axiom,
    ! [K2: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_7102_fact__numeral,axiom,
    ! [K2: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(K2)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_7103_fact__numeral,axiom,
    ! [K2: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_7104_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_7105_square__fact__le__2__fact,axiom,
    ! [Nb: 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,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_7106_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_7107_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7108_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7109_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7110_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7111_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7112_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7113_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7114_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7115_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7116_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),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_7117_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X4)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X4)) ) ) ).

% lemma_tan_total
tff(fact_7118_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X4)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X4) = 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),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),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X4 ) ) ) ).

% tan_total
tff(fact_7119_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),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),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_7120_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),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),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_7121_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),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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,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_7122_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X4)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_7123_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_7124_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_7125_add__tan__eq,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(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_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))) ) ) ) ).

% add_tan_eq
tff(fact_7126_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_7127_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),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_7128_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_7129_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)),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_7130_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),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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,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_7131_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),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),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_7132_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),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_7133_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_7134_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),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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_7135_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),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),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_7136_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_nn(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_7137_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_no(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_7138_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_rat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_np(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_7139_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_nq(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_7140_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_int_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_nr(Xb),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_7141_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ns(H),J),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_7142_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_7143_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_7144_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_7145_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_7146_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_7147_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_7148_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),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),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_7149_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_7150_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,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,numeral_numeral_real(aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_7151_cos__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(cos_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),aa_real_real(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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X)),zero_zero_real) ).

% cos_coeff_def
tff(fact_7152_cos__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_nt,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_paired
tff(fact_7153_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nu,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_7154_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nu,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_7155_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nu,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_7156_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] : aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nu,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_7157_sin__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(sin_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),zero_zero_real,aa_real_real(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,X),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X))) ).

% sin_coeff_def
tff(fact_7158_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T3))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nv,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7159_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_7160_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_7161_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,Xb) = one_one_real )
    <=> ( Xb = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_7162_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_7163_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_7164_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_7165_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_7166_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_7167_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_7168_exp__times__arg__commute,axiom,
    ! [A4: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A4)),A4) = aa_complex_complex(times_times_complex(A4),aa_complex_complex(exp_complex,A4)) ).

% exp_times_arg_commute
tff(fact_7169_exp__times__arg__commute,axiom,
    ! [A4: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A4)),A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A4),aa_real_real(exp_real,A4)) ).

% exp_times_arg_commute
tff(fact_7170_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_7171_exp__add__commuting,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_7172_exp__add__commuting,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_7173_mult__exp__exp,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).

% mult_exp_exp
tff(fact_7174_mult__exp__exp,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).

% mult_exp_exp
tff(fact_7175_exp__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_7176_exp__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_7177_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_7178_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_7179_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_7180_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_7181_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_7182_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_7183_exp__of__nat2__mult,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_7184_exp__of__nat2__mult,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_7185_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),Xb)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_7186_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_7187_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_7188_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_7189_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real))
          & ( aa_real_real(exp_real,X4) = Y ) ) ) ).

% lemma_exp_total
tff(fact_7190_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_7191_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_7192_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one))) ).

% exp_le
tff(fact_7193_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7194_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7195_tanh__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_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(exp_real,aa_real_real(uminus_uminus_real,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% tanh_altdef
tff(fact_7196_tanh__altdef,axiom,
    ! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).

% tanh_altdef
tff(fact_7197_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_7198_exp__double,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_7199_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_7200_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_7201_exp__bound__half,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),numeral_numeral_real(aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7202_exp__bound__half,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),numeral_numeral_real(aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7203_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_7204_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),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,numeral_numeral_real(aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_7205_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_7206_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7207_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7208_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_7209_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_7210_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nv,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_7211_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)))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7212_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_7213_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),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_7214_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),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_7215_pochhammer__double,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7216_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7217_pochhammer__double,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7218_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_nw,Nb),zero_zero_complex) ).

% of_nat_code
tff(fact_7219_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_nx,Nb),zero_zero_int) ).

% of_nat_code
tff(fact_7220_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_ny),Nb),zero_zero_real) ).

% of_nat_code
tff(fact_7221_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_nz),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_7222_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = semiri7787848453975740701ux_rat(aTP_Lamm_oa,Nb,zero_zero_rat) ).

% of_nat_code
tff(fact_7223_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_ob(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_7224_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_oc(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_7225_gchoose__row__sum__weighted,axiom,
    ! [R2: real,Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_od,R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = 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,Mb))),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_7226_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_7227_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_7228_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K2)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_7229_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K2)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_7230_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K2)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_7231_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_7232_gbinomial__0_I2_J,axiom,
    ! [K2: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K2)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_7233_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_7234_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_7235_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_7236_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_7237_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_7238_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_7239_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_7240_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_7241_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_7242_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_7243_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_7244_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_7245_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_7246_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_7247_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs2
tff(fact_7248_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_7249_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_7250_real__sqrt__abs,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs
tff(fact_7251_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_7252_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_7253_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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_squared_eq
tff(fact_7254_real__sqrt__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_7255_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_7256_pochhammer__pos,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7257_pochhammer__pos,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7258_pochhammer__pos,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7259_pochhammer__pos,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_7260_pochhammer__neq__0__mono,axiom,
    ! [A: complex,Mb: nat,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7261_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7262_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7263_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7264_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7265_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7266_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),Nb) ).

% pochhammer_fact
tff(fact_7267_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Nb) ).

% pochhammer_fact
tff(fact_7268_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),Nb) ).

% pochhammer_fact
tff(fact_7269_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),Nb) ).

% pochhammer_fact
tff(fact_7270_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Nb) ).

% pochhammer_fact
tff(fact_7271_gbinomial__pochhammer,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K2))),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer
tff(fact_7272_gbinomial__pochhammer,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(A)),K2)),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer
tff(fact_7273_gbinomial__pochhammer,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = 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)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K2))),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer
tff(fact_7274_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer'
tff(fact_7275_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer'
tff(fact_7276_gbinomial__pochhammer_H,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = 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,A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer'
tff(fact_7277_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_7278_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_7279_gbinomial__Suc__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_7280_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_7281_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K2) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7282_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K2) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7283_pochhammer__nonneg,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7284_pochhammer__nonneg,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7285_pochhammer__nonneg,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7286_pochhammer__nonneg,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_7287_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_7288_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_7289_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_7290_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_7291_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_7292_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_7293_gbinomial__addition__formula,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_addition_formula
tff(fact_7294_gbinomial__addition__formula,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)) = 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,A),one_one_real)),aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K2)) ).

% gbinomial_addition_formula
tff(fact_7295_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K2)) ).

% gbinomial_addition_formula
tff(fact_7296_gbinomial__absorb__comp,axiom,
    ! [A: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(gbinomial_complex(A),K2)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorb_comp
tff(fact_7297_gbinomial__absorb__comp,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K2)) ).

% gbinomial_absorb_comp
tff(fact_7298_gbinomial__absorb__comp,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K2)) ).

% gbinomial_absorb_comp
tff(fact_7299_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K2)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7300_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(gbinomial_rat(A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7301_gbinomial__mult__1,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_7302_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_7303_gbinomial__mult__1_H,axiom,
    ! [A: real,K2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_7304_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_7305_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_7306_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_7307_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_7308_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_7309_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_7310_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_7311_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_7312_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_7313_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_7314_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)) ).

% pochhammer_rec'
tff(fact_7315_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)) ).

% pochhammer_rec'
tff(fact_7316_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_7317_pochhammer__rec_H,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_7318_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7319_pochhammer__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7320_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7321_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7322_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7323_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7324_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7325_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7326_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7327_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7328_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7329_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7330_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7331_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7332_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7333_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7334_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7335_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7336_pochhammer__product_H,axiom,
    ! [Z: int,Nb: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7337_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,Mb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7338_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7339_pochhammer__product_H,axiom,
    ! [Z: rat,Nb: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_7340_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_7341_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_7342_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_7343_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K2)) ).

% Suc_times_gbinomial
tff(fact_7344_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% Suc_times_gbinomial
tff(fact_7345_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K2)) ).

% Suc_times_gbinomial
tff(fact_7346_gbinomial__absorption,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorption
tff(fact_7347_gbinomial__absorption,axiom,
    ! [K2: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K2)) ).

% gbinomial_absorption
tff(fact_7348_gbinomial__absorption,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K2)) ).

% gbinomial_absorption
tff(fact_7349_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Mb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7350_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Mb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7351_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Mb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7352_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7353_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7354_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_7355_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_7356_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_7357_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_7358_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_7359_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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))))) = Y )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_7360_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_7361_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_7362_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_7363_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_7364_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_45
tff(fact_7365_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_45
tff(fact_7366_gbinomial__factors,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))),aa_nat_complex(gbinomial_complex(A),K2)) ).

% gbinomial_factors
tff(fact_7367_gbinomial__factors,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% gbinomial_factors
tff(fact_7368_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))),aa_nat_rat(gbinomial_rat(A),K2)) ).

% gbinomial_factors
tff(fact_7369_gbinomial__rec,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K2)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_7370_gbinomial__rec,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_7371_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K2)),divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_7372_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_7373_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),Nb)) ).

% gbinomial_index_swap
tff(fact_7374_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: 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)),K2)),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,Nb))),one_one_real)),K2)) = 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)),Nb)),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,K2))),one_one_real)),Nb)) ).

% gbinomial_index_swap
tff(fact_7375_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)),one_one_rat)),Nb)) ).

% gbinomial_index_swap
tff(fact_7376_gbinomial__negated__upper,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K2)),A)),one_one_complex)),K2)) ).

% gbinomial_negated_upper
tff(fact_7377_gbinomial__negated__upper,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = 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)),K2)),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,K2)),A)),one_one_real)),K2)) ).

% gbinomial_negated_upper
tff(fact_7378_gbinomial__negated__upper,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(minus_minus_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K2),A),one_one_rat)),K2)) ).

% gbinomial_negated_upper
tff(fact_7379_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K2)) = 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)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7380_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7381_pochhammer__absorb__comp,axiom,
    ! [R2: int,K2: 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,K2))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K2)) = 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)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7382_pochhammer__absorb__comp,axiom,
    ! [R2: real,K2: 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,K2))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K2)) = 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)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7383_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K2: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(R2,aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(R2)),K2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(R2)),one_one_rat)),K2)) ).

% pochhammer_absorb_comp
tff(fact_7384_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% pochhammer_same
tff(fact_7385_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% pochhammer_same
tff(fact_7386_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% pochhammer_same
tff(fact_7387_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% pochhammer_same
tff(fact_7388_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Nb) = aa_real_real(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)),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% pochhammer_same
tff(fact_7389_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_7390_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_7391_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_7392_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_7393_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_7394_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_7395_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)),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_7396_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_30
tff(fact_7397_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_60
tff(fact_7398_gbinomial__minus,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% gbinomial_minus
tff(fact_7399_gbinomial__minus,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K2) = 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)),K2)),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,A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% gbinomial_minus
tff(fact_7400_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(uminus_uminus_rat(A)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K2)),one_one_rat)),K2)) ).

% gbinomial_minus
tff(fact_7401_complex__norm,axiom,
    ! [Xb: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(aa_rea2649881322057434177omplex(complex2,Xb),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))))) ).

% complex_norm
tff(fact_7402_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7403_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = 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,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7404_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7405_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% pochhammer_minus
tff(fact_7406_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2)) ).

% pochhammer_minus
tff(fact_7407_pochhammer__minus,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2) = 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)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2)) ).

% pochhammer_minus
tff(fact_7408_pochhammer__minus,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2) = 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)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% pochhammer_minus
tff(fact_7409_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% pochhammer_minus
tff(fact_7410_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K2)) ).

% pochhammer_minus'
tff(fact_7411_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2)) ).

% pochhammer_minus'
tff(fact_7412_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2) = 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)),K2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K2)) ).

% pochhammer_minus'
tff(fact_7413_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2) = 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)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2)) ).

% pochhammer_minus'
tff(fact_7414_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K2)) ).

% pochhammer_minus'
tff(fact_7415_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_7416_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_7417_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_7418_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)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_7419_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_7420_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : groups2073611262835488442omplex(aTP_Lamm_oe(K2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7421_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_of(K2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7422_gbinomial__sum__up__index,axiom,
    ! [K2: nat,Nb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_og,K2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_7423_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_7424_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,numeral_numeral_real(aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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_7425_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_7426_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7427_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7428_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_complex(gbinomial_complex(A),K2) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7429_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7430_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7431_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),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),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_7432_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_7433_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_7434_fact__double,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_double
tff(fact_7435_fact__double,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_double
tff(fact_7436_fact__double,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_double
tff(fact_7437_gbinomial__code,axiom,
    ! [A: complex,K2: nat] :
      aa_nat_complex(gbinomial_complex(A),K2) = $ite(K2 = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_oh(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K2))) ).

% gbinomial_code
tff(fact_7438_gbinomial__code,axiom,
    ! [A: rat,K2: nat] :
      aa_nat_rat(gbinomial_rat(A),K2) = $ite(K2 = zero_zero_nat,one_one_rat,divide_divide_rat(set_fo1949268297981939178at_rat(aTP_Lamm_oi(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat),one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,K2))) ).

% gbinomial_code
tff(fact_7439_gbinomial__code,axiom,
    ! [A: real,K2: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = $ite(K2 = 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_oj(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K2))) ).

% gbinomial_code
tff(fact_7440_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups6464643781859351333omplex(aTP_Lamm_ok(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7441_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: 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,Nb))),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),numeral_numeral_real(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_ol,Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7442_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_om(Z),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7443_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_on(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_7444_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_oo(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_7445_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_op(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_7446_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,set_fo1949268297981939178at_rat(aTP_Lamm_oq(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),one_one_rat)) ).

% pochhammer_code
tff(fact_7447_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_or(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_7448_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_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(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_7449_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_ob(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_7450_gbinomial__partial__row__sum,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_oc(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_7451_gbinomial__partial__row__sum,axiom,
    ! [A: real,Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_od,A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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,Mb)),one_one_real)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_7452_atMost__iff,axiom,
    ! [I: real,K2: real] :
      ( member_real(I,set_ord_atMost_real(K2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),K2) ) ).

% atMost_iff
tff(fact_7453_atMost__iff,axiom,
    ! [I: set_nat,K2: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(I),K2) ) ).

% atMost_iff
tff(fact_7454_atMost__iff,axiom,
    ! [I: rat,K2: rat] :
      ( member_rat(I,set_ord_atMost_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),K2) ) ).

% atMost_iff
tff(fact_7455_atMost__iff,axiom,
    ! [I: num,K2: num] :
      ( member_num(I,set_ord_atMost_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),K2) ) ).

% atMost_iff
tff(fact_7456_atMost__iff,axiom,
    ! [I: int,K2: int] :
      ( member_int(I,set_ord_atMost_int(K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K2) ) ).

% atMost_iff
tff(fact_7457_atMost__iff,axiom,
    ! [I: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2) ) ).

% atMost_iff
tff(fact_7458_prod_Oneutral__const,axiom,
    ! [A4: set_nat] : groups708209901874060359at_nat(aTP_Lamm_os,A4) = one_one_nat ).

% prod.neutral_const
tff(fact_7459_prod_Oneutral__const,axiom,
    ! [A4: set_nat] : groups705719431365010083at_int(aTP_Lamm_ot,A4) = one_one_int ).

% prod.neutral_const
tff(fact_7460_prod_Oneutral__const,axiom,
    ! [A4: set_int] : groups1705073143266064639nt_int(aTP_Lamm_ou,A4) = one_one_int ).

% prod.neutral_const
tff(fact_7461_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_7462_prod_Oempty,axiom,
    ! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_7463_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_7464_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_7465_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_7466_prod_Oempty,axiom,
    ! [G: fun_int_rat] : groups1072433553688619179nt_rat(G,bot_bot_set_int) = one_one_rat ).

% prod.empty
tff(fact_7467_prod_Oempty,axiom,
    ! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).

% prod.empty
tff(fact_7468_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_7469_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_7470_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_7471_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_complex] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups6464643781859351333omplex(G,A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7472_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( ~ finite3207457112153483333omplex(A4)
     => ( groups3708469109370488835omplex(G,A4) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7473_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_real] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups129246275422532515t_real(G,A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_7474_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( ~ finite3207457112153483333omplex(A4)
     => ( groups766887009212190081x_real(G,A4) = one_one_real ) ) ).

% prod.infinite
tff(fact_7475_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups73079841787564623at_rat(G,A4) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7476_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( ~ finite3207457112153483333omplex(A4)
     => ( groups225925009352817453ex_rat(G,A4) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7477_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_nat] :
      ( ~ finite3207457112153483333omplex(A4)
     => ( groups861055069439313189ex_nat(G,A4) = one_one_nat ) ) ).

% prod.infinite
tff(fact_7478_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_int] :
      ( ~ finite3207457112153483333omplex(A4)
     => ( groups858564598930262913ex_int(G,A4) = one_one_int ) ) ).

% prod.infinite
tff(fact_7479_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups708209901874060359at_nat(G,A4) = one_one_nat ) ) ).

% prod.infinite
tff(fact_7480_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups705719431365010083at_int(G,A4) = one_one_int ) ) ).

% prod.infinite
tff(fact_7481_atMost__subset__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(Xb),set_or4236626031148496127et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).

% atMost_subset_iff
tff(fact_7482_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_7483_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_7484_atMost__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( 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_7485_atMost__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_7486_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( finite_finite_int(S3)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ov(A),B),S3) = $ite(member_int(A,S3),aa_int_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7487_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_ow(A),B),S3) = $ite(member_real(A,S3),aa_real_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7488_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ox(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7489_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( finite3207457112153483333omplex(S3)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_oy(A),B),S3) = $ite(member_complex(A,S3),aa_complex_complex(B,A),one_one_complex) ) ) ).

% prod.delta
tff(fact_7490_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( finite_finite_int(S3)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_oz(A),B),S3) = $ite(member_int(A,S3),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7491_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pa(A),B),S3) = $ite(member_real(A,S3),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7492_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pb(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7493_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( finite3207457112153483333omplex(S3)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pc(A),B),S3) = $ite(member_complex(A,S3),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta
tff(fact_7494_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S3)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pd(A),B),S3) = $ite(member_int(A,S3),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_7495_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_pe(A),B),S3) = $ite(member_real(A,S3),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta
tff(fact_7496_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( finite_finite_int(S3)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_pf(A),B),S3) = $ite(member_int(A,S3),aa_int_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7497_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_pg(A),B),S3) = $ite(member_real(A,S3),aa_real_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7498_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ph(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7499_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( finite3207457112153483333omplex(S3)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_pi(A),B),S3) = $ite(member_complex(A,S3),aa_complex_complex(B,A),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7500_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( finite_finite_int(S3)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_pj(A),B),S3) = $ite(member_int(A,S3),aa_int_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7501_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pk(A),B),S3) = $ite(member_real(A,S3),aa_real_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7502_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pl(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7503_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( finite3207457112153483333omplex(S3)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_pm(A),B),S3) = $ite(member_complex(A,S3),aa_complex_real(B,A),one_one_real) ) ) ).

% prod.delta'
tff(fact_7504_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S3)
     => ( groups1072433553688619179nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pn(A),B),S3) = $ite(member_int(A,S3),aa_int_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7505_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_po(A),B),S3) = $ite(member_real(A,S3),aa_real_rat(B,A),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7506_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H2: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H),set_or4236626031148496127et_nat(H2))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
        | aa_set_nat_o(ord_less_eq_set_nat(H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7507_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_7508_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_7509_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7510_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( ord_less_eq_set_int(set_or1266510415728281911st_int(L,H),set_ord_atMost_int(H2))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_7511_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_7512_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_7513_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_7514_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_7515_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_7516_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7517_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7518_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7519_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_7520_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_7521_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_7522_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_7523_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_7524_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7525_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7526_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7527_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7528_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_7529_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A4: set_nat] :
      ( ( groups6464643781859351333omplex(G,A4) != one_one_complex )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ( aa_nat_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7530_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A4: set_int] :
      ( ( groups7440179247065528705omplex(G,A4) != one_one_complex )
     => ~ ! [A3: int] :
            ( member_int(A3,A4)
           => ( aa_int_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7531_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A4: set_real] :
      ( ( groups713298508707869441omplex(G,A4) != one_one_complex )
     => ~ ! [A3: real] :
            ( member_real(A3,A4)
           => ( aa_real_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7532_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_complex,A4: set_complex] :
      ( ( groups3708469109370488835omplex(G,A4) != one_one_complex )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A4)
           => ( aa_complex_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7533_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A4: set_nat] :
      ( ( groups129246275422532515t_real(G,A4) != one_one_real )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ( aa_nat_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7534_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A4: set_int] :
      ( ( groups2316167850115554303t_real(G,A4) != one_one_real )
     => ~ ! [A3: int] :
            ( member_int(A3,A4)
           => ( aa_int_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7535_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A4: set_real] :
      ( ( groups1681761925125756287l_real(G,A4) != one_one_real )
     => ~ ! [A3: real] :
            ( member_real(A3,A4)
           => ( aa_real_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7536_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A4: set_complex] :
      ( ( groups766887009212190081x_real(G,A4) != one_one_real )
     => ~ ! [A3: complex] :
            ( member_complex(A3,A4)
           => ( aa_complex_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7537_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A4: set_nat] :
      ( ( groups73079841787564623at_rat(G,A4) != one_one_rat )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ( aa_nat_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7538_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A4: set_int] :
      ( ( groups1072433553688619179nt_rat(G,A4) != one_one_rat )
     => ~ ! [A3: int] :
            ( member_int(A3,A4)
           => ( aa_int_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7539_prod_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ( aa_nat_nat(G,X4) = one_one_nat ) )
     => ( groups708209901874060359at_nat(G,A4) = one_one_nat ) ) ).

% prod.neutral
tff(fact_7540_prod_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ( aa_nat_int(G,X4) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A4) = one_one_int ) ) ).

% prod.neutral
tff(fact_7541_prod_Oneutral,axiom,
    ! [A4: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => ( aa_int_int(G,X4) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A4) = one_one_int ) ) ).

% prod.neutral
tff(fact_7542_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A4: set_nat] : groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pp(G),H),A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,A4)),groups708209901874060359at_nat(H,A4)) ).

% prod.distrib
tff(fact_7543_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H: fun_nat_int,A4: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_pq(G),H),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,A4)),groups705719431365010083at_int(H,A4)) ).

% prod.distrib
tff(fact_7544_prod_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A4: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pr(G),H),A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A4)),groups1705073143266064639nt_int(H,A4)) ).

% prod.distrib
tff(fact_7545_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,groups708209901874060359at_nat(F,A4)),Nb) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ps(F),Nb),A4) ).

% prod_power_distrib
tff(fact_7546_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A4: set_nat,Nb: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A4)),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_pt(F),Nb),A4) ).

% prod_power_distrib
tff(fact_7547_prod__power__distrib,axiom,
    ! [F: fun_int_int,A4: set_int,Nb: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A4)),Nb) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_pu(F),Nb),A4) ).

% prod_power_distrib
tff(fact_7548_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gh(F),A),A4)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(F,A4)),A) ).

% mod_prod_eq
tff(fact_7549_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A4: set_nat] : aa_int_int(modulo_modulo_int(groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_pv(F),A),A4)),A) = aa_int_int(modulo_modulo_int(groups705719431365010083at_int(F,A4)),A) ).

% mod_prod_eq
tff(fact_7550_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A4: set_int] : aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_gg(F),A),A4)),A) = aa_int_int(modulo_modulo_int(groups1705073143266064639nt_int(F,A4)),A) ).

% mod_prod_eq
tff(fact_7551_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_7552_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_7553_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_hy(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_7554_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_7555_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(aTP_Lamm_pw(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_py(A),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_7556_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,Nb: nat] : groups705719431365010083at_int(aTP_Lamm_pz(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qb(A),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_7557_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_qc,U)) ).

% atMost_def
tff(fact_7558_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_qd(U)) ).

% atMost_def
tff(fact_7559_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_qe,U)) ).

% atMost_def
tff(fact_7560_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_qf,U)) ).

% atMost_def
tff(fact_7561_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_qg,U)) ).

% atMost_def
tff(fact_7562_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bt,U)) ).

% atMost_def
tff(fact_7563_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A4)
         => ( 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,groups129246275422532515t_real(F,A4)),groups129246275422532515t_real(G,A4)) ) ).

% prod_mono
tff(fact_7564_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( member_int(I2,A4)
         => ( 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,A4)),groups2316167850115554303t_real(G,A4)) ) ).

% prod_mono
tff(fact_7565_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,A4)
         => ( 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,A4)),groups1681761925125756287l_real(G,A4)) ) ).

% prod_mono
tff(fact_7566_prod__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I2)),aa_complex_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(G,A4)) ) ).

% prod_mono
tff(fact_7567_prod__mono,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A4)
         => ( 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,A4)),groups73079841787564623at_rat(G,A4)) ) ).

% prod_mono
tff(fact_7568_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,A4)
         => ( 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,A4)),groups1072433553688619179nt_rat(G,A4)) ) ).

% prod_mono
tff(fact_7569_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,A4)
         => ( 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,A4)),groups4061424788464935467al_rat(G,A4)) ) ).

% prod_mono
tff(fact_7570_prod__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(G,A4)) ) ).

% prod_mono
tff(fact_7571_prod__mono,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( member_int(I2,A4)
         => ( 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,A4)),groups1707563613775114915nt_nat(G,A4)) ) ).

% prod_mono
tff(fact_7572_prod__mono,axiom,
    ! [A4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,A4)
         => ( 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,A4)),groups4696554848551431203al_nat(G,A4)) ) ).

% prod_mono
tff(fact_7573_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups708209901874060359at_nat(F,A4)) ) ).

% prod_nonneg
tff(fact_7574_prod__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A4)) ) ).

% prod_nonneg
tff(fact_7575_prod__nonneg,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A4)) ) ).

% prod_nonneg
tff(fact_7576_prod__pos,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A4)) ) ).

% prod_pos
tff(fact_7577_prod__pos,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A4)) ) ).

% prod_pos
tff(fact_7578_prod__pos,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X4)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A4)) ) ).

% prod_pos
tff(fact_7579_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A4)) ) ).

% prod_ge_1
tff(fact_7580_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A4)) ) ).

% prod_ge_1
tff(fact_7581_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A4)) ) ).

% prod_ge_1
tff(fact_7582_prod__ge__1,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A4)) ) ).

% prod_ge_1
tff(fact_7583_prod__ge__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7584_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7585_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7586_prod__ge__1,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X4)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A4)) ) ).

% prod_ge_1
tff(fact_7587_prod__ge__1,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A4)) ) ).

% prod_ge_1
tff(fact_7588_prod__ge__1,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A4)) ) ).

% prod_ge_1
tff(fact_7589_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_qh(F),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_7590_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups129246275422532515t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_qi(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_7591_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_qj(F),A,B,one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_7592_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : groups708209901874060359at_nat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_qk(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_7593_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ql(F),A,B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_7594_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_7595_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_7596_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_hy(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_7597_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_7598_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_ord_atMost_nat,K2) ).

% lessThan_Suc_atMost
tff(fact_7599_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_complex,P: fun_int_o] :
      ( finite_finite_int(A4)
     => ( groups7440179247065528705omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_qm(A4),P))) = groups7440179247065528705omplex(aa_fun6439487526662521568omplex(aTP_Lamm_qn(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7600_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_complex,P: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups713298508707869441omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_qo(A4),P))) = groups713298508707869441omplex(aa_fun1098856492951973088omplex(aTP_Lamm_qp(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7601_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_complex,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups6464643781859351333omplex(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_qq(A4),P))) = groups6464643781859351333omplex(aa_fun1400689748890728928omplex(aTP_Lamm_qr(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7602_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_complex,P: fun_complex_o] :
      ( finite3207457112153483333omplex(A4)
     => ( groups3708469109370488835omplex(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_qs(A4),P))) = groups3708469109370488835omplex(aa_fun3681620399087435360omplex(aTP_Lamm_qt(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7603_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_real,P: fun_int_o] :
      ( finite_finite_int(A4)
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_qm(A4),P))) = groups2316167850115554303t_real(aa_fun7506669883404309598t_real(aTP_Lamm_qu(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7604_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_real,P: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_qo(A4),P))) = groups1681761925125756287l_real(aa_fun4448911596306054750l_real(aTP_Lamm_qv(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7605_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_real,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups129246275422532515t_real(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_qq(A4),P))) = groups129246275422532515t_real(aa_fun5510454223943551838t_real(aTP_Lamm_qw(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7606_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( finite3207457112153483333omplex(A4)
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_qs(A4),P))) = groups766887009212190081x_real(aa_fun8261057609414086622x_real(aTP_Lamm_qx(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7607_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_rat,P: fun_int_o] :
      ( finite_finite_int(A4)
     => ( groups1072433553688619179nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_qm(A4),P))) = groups1072433553688619179nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_qy(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7608_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_rat,P: fun_real_o] :
      ( finite_finite_real(A4)
     => ( groups4061424788464935467al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_qo(A4),P))) = groups4061424788464935467al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_qz(G),P),A4) ) ) ).

% prod.inter_filter
tff(fact_7609_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups708209901874060359at_nat(aTP_Lamm_hy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7610_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7611_power__sum,axiom,
    ! [C: real,F: fun_nat_nat,A4: set_nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_ra(C),F),A4) ).

% power_sum
tff(fact_7612_power__sum,axiom,
    ! [C: complex,F: fun_nat_nat,A4: set_nat] : aa_nat_complex(power_power_complex(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_rb(C),F),A4) ).

% power_sum
tff(fact_7613_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rc(C),F),A4) ).

% power_sum
tff(fact_7614_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A4: set_nat] : aa_nat_int(power_power_int(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_rd(C),F),A4) ).

% power_sum
tff(fact_7615_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A4: set_int] : aa_nat_int(power_power_int(C),groups4541462559716669496nt_nat(F,A4)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_re(C),F),A4) ).

% power_sum
tff(fact_7616_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K2: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ia(G),K2),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7617_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,K2: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_rf(G),K2),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7618_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X4)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7619_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X4)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7620_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7621_prod__le__1,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X4)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),one_one_real) ) ).

% prod_le_1
tff(fact_7622_prod__le__1,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7623_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7624_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X4)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7625_prod__le__1,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( member_complex(X4,A4)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X4)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),one_one_rat) ) ).

% prod_le_1
tff(fact_7626_prod__le__1,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( member_int(X4,A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A4)),one_one_nat) ) ).

% prod_le_1
tff(fact_7627_prod__le__1,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A4)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A4)),one_one_nat) ) ).

% prod_le_1
tff(fact_7628_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups6464643781859351333omplex(H,S3)),groups6464643781859351333omplex(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7629_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X22)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X15),Y15)),aa_complex_complex(times_times_complex(X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(H,X4)),aa_complex_complex(G,X4)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups3708469109370488835omplex(H,S3)),groups3708469109370488835omplex(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7630_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: 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)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_nat_real(H,X4)),aa_nat_real(G,X4)) )
           => aa_real_o(aa_real_fun_real_o(R,groups129246275422532515t_real(H,S3)),groups129246275422532515t_real(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7631_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: 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)
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X22)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X4)),aa_complex_real(G,X4)) )
           => aa_real_o(aa_real_fun_real_o(R,groups766887009212190081x_real(H,S3)),groups766887009212190081x_real(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7632_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: 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)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups73079841787564623at_rat(H,S3)),groups73079841787564623at_rat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7633_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: 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)
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X22)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X15),Y15)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups225925009352817453ex_rat(H,S3)),groups225925009352817453ex_rat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7634_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups861055069439313189ex_nat(H,S3)),groups861055069439313189ex_nat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7635_prod_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_int_o(aa_int_fun_int_o(R,one_one_int),one_one_int)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(times_times_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(times_times_int,X22),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_complex_int(H,X4)),aa_complex_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups858564598930262913ex_int(H,S3)),groups858564598930262913ex_int(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7636_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_nat,H: fun_nat_nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X22)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(H,X4)),aa_nat_nat(G,X4)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups708209901874060359at_nat(H,S3)),groups708209901874060359at_nat(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7637_prod_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,one_one_int),one_one_int)
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X22)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(times_times_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(times_times_int,X22),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X4)),aa_nat_int(G,X4)) )
           => aa_int_o(aa_int_fun_int_o(R,groups705719431365010083at_int(H,S3)),groups705719431365010083at_int(G,S3)) ) ) ) ) ).

% prod.related
tff(fact_7638_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(S4)
     => ( finite_finite_int(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_int(aa_int_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_int(aa_int_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7639_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( finite_finite_int(S4)
     => ( finite_finite_real(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T,T2)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T2))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T2))
                   => member_int(aa_real_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T2)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7640_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_int,S3: 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)
     => ( finite_finite_int(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_int(aa_real_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_real(aa_int_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7641_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_real,S3: 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(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T,T2)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T2))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T2))
                   => member_real(aa_real_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T2)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7642_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_complex,S3: set_int,I: fun_complex_int,J: fun_int_complex,T: set_complex,G: fun_int_complex,H: fun_complex_complex] :
      ( finite_finite_int(S4)
     => ( finite3207457112153483333omplex(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_complex(aa_int_complex(J,A3),minus_811609699411566653omplex(T,T2)) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                   => member_int(aa_complex_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( member_complex(B3,T2)
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_complex_complex(H,aa_int_complex(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7643_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T2: set_complex,S3: 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)
     => ( finite3207457112153483333omplex(T2)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_complex(aa_real_complex(J,A3),minus_811609699411566653omplex(T,T2)) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                   => member_real(aa_complex_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( member_complex(B3,T2)
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_complex_complex(H,aa_real_complex(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7644_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T2: set_int,S3: set_complex,I: fun_int_complex,J: fun_complex_int,T: set_int,G: fun_complex_complex,H: fun_int_complex] :
      ( finite3207457112153483333omplex(S4)
     => ( finite_finite_int(T2)
       => ( ! [A3: complex] :
              ( member_complex(A3,minus_811609699411566653omplex(S3,S4))
             => ( aa_int_complex(I,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(S3,S4))
               => member_int(aa_complex_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_complex(aa_int_complex(I,B3),minus_811609699411566653omplex(S3,S4)) )
               => ( ! [A3: complex] :
                      ( member_complex(A3,S4)
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( member_complex(A3,S3)
                         => ( aa_int_complex(H,aa_complex_int(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S3) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7645_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T2: set_real,S3: set_complex,I: fun_real_complex,J: fun_complex_real,T: set_real,G: fun_complex_complex,H: fun_real_complex] :
      ( finite3207457112153483333omplex(S4)
     => ( finite_finite_real(T2)
       => ( ! [A3: complex] :
              ( member_complex(A3,minus_811609699411566653omplex(S3,S4))
             => ( aa_real_complex(I,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(S3,S4))
               => member_real(aa_complex_real(J,A3),minus_minus_set_real(T,T2)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T,T2))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T,T2))
                   => member_complex(aa_real_complex(I,B3),minus_811609699411566653omplex(S3,S4)) )
               => ( ! [A3: complex] :
                      ( member_complex(A3,S4)
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T2)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( member_complex(A3,S3)
                         => ( aa_real_complex(H,aa_complex_real(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S3) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7646_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T2: set_complex,S3: set_complex,I: fun_complex_complex,J: fun_complex_complex,T: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( finite3207457112153483333omplex(S4)
     => ( finite3207457112153483333omplex(T2)
       => ( ! [A3: complex] :
              ( member_complex(A3,minus_811609699411566653omplex(S3,S4))
             => ( aa_complex_complex(I,aa_complex_complex(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(S3,S4))
               => member_complex(aa_complex_complex(J,A3),minus_811609699411566653omplex(T,T2)) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                 => ( aa_complex_complex(J,aa_complex_complex(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( member_complex(B3,minus_811609699411566653omplex(T,T2))
                   => member_complex(aa_complex_complex(I,B3),minus_811609699411566653omplex(S3,S4)) )
               => ( ! [A3: complex] :
                      ( member_complex(A3,S4)
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( member_complex(B3,T2)
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( member_complex(A3,S3)
                         => ( aa_complex_complex(H,aa_complex_complex(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7647_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T2: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(S4)
     => ( finite_finite_int(T2)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_int(aa_int_int(J,A3),minus_minus_set_int(T,T2)) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(T,T2))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( member_int(B3,minus_minus_set_int(T,T2))
                   => member_int(aa_int_int(I,B3),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_real(G,A3) = one_one_real ) )
                 => ( ! [B3: int] :
                        ( member_int(B3,T2)
                       => ( aa_int_real(H,B3) = one_one_real ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_int_real(H,aa_int_int(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7648_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups129246275422532515t_real(aTP_Lamm_rg(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_7649_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_rh(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_7650_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups708209901874060359at_nat(aTP_Lamm_ri(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_7651_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_rj(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_7652_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_complex] :
      ( finite_finite_real(A4)
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_rk(G)))) = groups713298508707869441omplex(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7653_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( finite3207457112153483333omplex(A4)
     => ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rl(G)))) = groups3708469109370488835omplex(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7654_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( groups1681761925125756287l_real(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_rm(G)))) = groups1681761925125756287l_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7655_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A4)
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rn(G)))) = groups766887009212190081x_real(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7656_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( groups4061424788464935467al_rat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_ro(G)))) = groups4061424788464935467al_rat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7657_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A4)
     => ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rp(G)))) = groups225925009352817453ex_rat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7658_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_nat] :
      ( finite_finite_real(A4)
     => ( groups4696554848551431203al_nat(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_rq(G)))) = groups4696554848551431203al_nat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7659_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rr(G)))) = groups861055069439313189ex_nat(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7660_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: fun_real_int] :
      ( finite_finite_real(A4)
     => ( groups4694064378042380927al_int(G,minus_minus_set_real(A4,collect_real(aTP_Lamm_rs(G)))) = groups4694064378042380927al_int(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7661_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(A4)
     => ( groups858564598930262913ex_int(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_rt(G)))) = groups858564598930262913ex_int(G,A4) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7662_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_7663_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ru(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_7664_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(A),set_ord_lessThan_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_7665_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(A),set_ord_lessThan_num(B))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_7666_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(set_ord_atMost_int(A),set_ord_lessThan_int(B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_7667_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_7668_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(set_ord_atMost_real(A),set_or5984915006950818249n_real(B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_7669_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ig(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_7670_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_rv(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_7671_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rw(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rx(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7672_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ry(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rz(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7673_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sa(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sb(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7674_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sc(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sd(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7675_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_se(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sf(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_7676_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_real] :
      ( finite_finite_int(I6)
     => ( member_int(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7677_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7678_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( 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),I6)
               => 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),groups129246275422532515t_real(F,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7679_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( member_complex(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7680_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: fun_int_rat] :
      ( finite_finite_int(I6)
     => ( member_int(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( member_int(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7681_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7682_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I),I6)
       => ( 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),I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7683_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( member_complex(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7684_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: fun_real_int] :
      ( finite_finite_real(I6)
     => ( member_real(I,I6)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7685_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I,I6)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I))
         => ( ! [I2: complex] :
                ( member_complex(I2,I6)
               => 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,I6)) ) ) ) ) ).

% less_1_prod2
tff(fact_7686_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7687_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => 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),groups129246275422532515t_real(F,I6)) ) ) ) ).

% less_1_prod
tff(fact_7688_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7689_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7690_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7691_less__1__prod,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7692_less__1__prod,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7693_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7694_less__1__prod,axiom,
    ! [I6: set_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7695_less__1__prod,axiom,
    ! [I6: set_real,F: fun_real_int] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => 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,I6)) ) ) ) ).

% less_1_prod
tff(fact_7696_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups766887009212190081x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A4,B4))),groups766887009212190081x_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7697_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups225925009352817453ex_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups225925009352817453ex_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7698_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups861055069439313189ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7699_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( ord_le211207098394363844omplex(B4,A4)
     => ( finite3207457112153483333omplex(A4)
       => ( groups858564598930262913ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A4,B4))),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7700_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups129246275422532515t_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,minus_minus_set_nat(A4,B4))),groups129246275422532515t_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7701_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups73079841787564623at_rat(G,A4) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,minus_minus_set_nat(A4,B4))),groups73079841787564623at_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7702_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups708209901874060359at_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,minus_minus_set_nat(A4,B4))),groups708209901874060359at_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7703_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups705719431365010083at_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,minus_minus_set_nat(A4,B4))),groups705719431365010083at_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7704_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: fun_int_int] :
      ( ord_less_eq_set_int(B4,A4)
     => ( finite_finite_int(A4)
       => ( groups1705073143266064639nt_int(G,A4) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,minus_minus_set_int(A4,B4))),groups1705073143266064639nt_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_7705_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
           => ( groups7440179247065528705omplex(G,T) = groups7440179247065528705omplex(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7706_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups713298508707869441omplex(G,T) = groups713298508707869441omplex(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7707_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,T) = groups3708469109370488835omplex(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7708_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups2316167850115554303t_real(G,T) = groups2316167850115554303t_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7709_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_real(G,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,T) = groups1681761925125756287l_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7710_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,T) = groups766887009212190081x_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7711_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups1072433553688619179nt_rat(G,T) = groups1072433553688619179nt_rat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7712_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups4061424788464935467al_rat(G,T) = groups4061424788464935467al_rat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7713_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups225925009352817453ex_rat(G,T) = groups225925009352817453ex_rat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7714_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_int,S3: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups1707563613775114915nt_nat(G,T) = groups1707563613775114915nt_nat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7715_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
           => ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7716_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_complex(G,X4) = aa_real_complex(H,X4) ) )
           => ( groups713298508707869441omplex(G,S3) = groups713298508707869441omplex(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7717_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7718_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_real(H,X4) = one_one_real ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7719_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_real(H,X4) = one_one_real ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_real(G,X4) = aa_real_real(H,X4) ) )
           => ( groups1681761925125756287l_real(G,S3) = groups1681761925125756287l_real(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7720_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(H,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7721_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups1072433553688619179nt_rat(G,S3) = groups1072433553688619179nt_rat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7722_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S3,T)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T,S3))
             => ( aa_real_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: real] :
                ( member_real(X4,S3)
               => ( aa_real_rat(G,X4) = aa_real_rat(H,X4) ) )
           => ( groups4061424788464935467al_rat(G,S3) = groups4061424788464935467al_rat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7723_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: complex] :
                ( member_complex(X4,S3)
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups225925009352817453ex_rat(G,S3) = groups225925009352817453ex_rat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7724_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_int,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(T)
     => ( ord_less_eq_set_int(S3,T)
       => ( ! [X4: int] :
              ( member_int(X4,minus_minus_set_int(T,S3))
             => ( aa_int_nat(H,X4) = one_one_nat ) )
         => ( ! [X4: int] :
                ( member_int(X4,S3)
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups1707563613775114915nt_nat(G,S3) = groups1707563613775114915nt_nat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7725_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T) = groups3708469109370488835omplex(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7726_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T) = groups766887009212190081x_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7727_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T) = groups225925009352817453ex_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7728_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T) = groups861055069439313189ex_nat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7729_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T) = groups858564598930262913ex_int(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7730_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,T) = groups6464643781859351333omplex(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7731_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( groups129246275422532515t_real(G,T) = groups129246275422532515t_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7732_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,T) = groups73079841787564623at_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7733_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_nat(G,X4) = one_one_nat ) )
         => ( groups708209901874060359at_nat(G,T) = groups708209901874060359at_nat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7734_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_int(G,X4) = one_one_int ) )
         => ( groups705719431365010083at_int(G,T) = groups705719431365010083at_int(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7735_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7736_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7737_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S3) = groups225925009352817453ex_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7738_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_nat(G,X4) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S3) = groups861055069439313189ex_nat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7739_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S3: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T)
     => ( ord_le211207098394363844omplex(S3,T)
       => ( ! [X4: complex] :
              ( member_complex(X4,minus_811609699411566653omplex(T,S3))
             => ( aa_complex_int(G,X4) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S3) = groups858564598930262913ex_int(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7740_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_complex(G,X4) = one_one_complex ) )
         => ( groups6464643781859351333omplex(G,S3) = groups6464643781859351333omplex(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7741_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_real(G,X4) = one_one_real ) )
         => ( groups129246275422532515t_real(G,S3) = groups129246275422532515t_real(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7742_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_rat(G,X4) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,S3) = groups73079841787564623at_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7743_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_nat(G,X4) = one_one_nat ) )
         => ( groups708209901874060359at_nat(G,S3) = groups708209901874060359at_nat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7744_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T)
       => ( ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(T,S3))
             => ( aa_nat_int(G,X4) = one_one_int ) )
         => ( groups705719431365010083at_int(G,S3) = groups705719431365010083at_int(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7745_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_complex(H,B3) = one_one_complex ) )
             => ( ( groups7440179247065528705omplex(G,C4) = groups7440179247065528705omplex(H,C4) )
               => ( groups7440179247065528705omplex(G,A4) = groups7440179247065528705omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7746_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) )
               => ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7747_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) )
               => ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7748_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_real(G,A3) = one_one_real ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_real(H,B3) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,C4) = groups2316167850115554303t_real(H,C4) )
               => ( groups2316167850115554303t_real(G,A4) = groups2316167850115554303t_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7749_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
               => ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7750_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) )
               => ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7751_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_rat(H,B3) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,C4) = groups1072433553688619179nt_rat(H,C4) )
               => ( groups1072433553688619179nt_rat(G,A4) = groups1072433553688619179nt_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7752_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) )
               => ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7753_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) )
               => ( groups225925009352817453ex_rat(G,A4) = groups225925009352817453ex_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7754_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_nat(H,B3) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,C4) = groups1707563613775114915nt_nat(H,C4) )
               => ( groups1707563613775114915nt_nat(G,A4) = groups1707563613775114915nt_nat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7755_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_complex(H,B3) = one_one_complex ) )
             => ( ( groups7440179247065528705omplex(G,A4) = groups7440179247065528705omplex(H,B4) )
              <=> ( groups7440179247065528705omplex(G,C4) = groups7440179247065528705omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7756_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A4) = groups713298508707869441omplex(H,B4) )
              <=> ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7757_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A4) = groups3708469109370488835omplex(H,B4) )
              <=> ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7758_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_real(G,A3) = one_one_real ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_real(H,B3) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,A4) = groups2316167850115554303t_real(H,B4) )
              <=> ( groups2316167850115554303t_real(G,C4) = groups2316167850115554303t_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7759_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A4) = groups1681761925125756287l_real(H,B4) )
              <=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7760_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A4) = groups766887009212190081x_real(H,B4) )
              <=> ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7761_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_rat(H,B3) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,A4) = groups1072433553688619179nt_rat(H,B4) )
              <=> ( groups1072433553688619179nt_rat(G,C4) = groups1072433553688619179nt_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7762_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A4,C4)
       => ( ord_less_eq_set_real(B4,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A4))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B4))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A4) = groups4061424788464935467al_rat(H,B4) )
              <=> ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7763_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(C4)
     => ( ord_le211207098394363844omplex(A4,C4)
       => ( ord_le211207098394363844omplex(B4,C4)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C4,A4))
               => ( aa_complex_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,minus_811609699411566653omplex(C4,B4))
                 => ( aa_complex_rat(H,B3) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,A4) = groups225925009352817453ex_rat(H,B4) )
              <=> ( groups225925009352817453ex_rat(G,C4) = groups225925009352817453ex_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7764_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( finite_finite_int(C4)
     => ( ord_less_eq_set_int(A4,C4)
       => ( ord_less_eq_set_int(B4,C4)
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(C4,A4))
               => ( aa_int_nat(G,A3) = one_one_nat ) )
           => ( ! [B3: int] :
                  ( member_int(B3,minus_minus_set_int(C4,B4))
                 => ( aa_int_nat(H,B3) = one_one_nat ) )
             => ( ( groups1707563613775114915nt_nat(G,A4) = groups1707563613775114915nt_nat(H,B4) )
              <=> ( groups1707563613775114915nt_nat(G,C4) = groups1707563613775114915nt_nat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7765_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7766_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7767_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7768_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7769_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7770_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7771_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7772_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7773_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7774_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,Mb)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7775_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7776_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7777_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7778_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7779_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_hy(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7780_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_7781_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),groups129246275422532515t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7782_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,Mb)),groups73079841787564623at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7783_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),groups708209901874060359at_nat(aTP_Lamm_hy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7784_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7785_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_7786_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_7787_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hy(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_7788_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_7789_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_kx(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_7790_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_ky(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_7791_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : groups6591440286371151544t_real(aTP_Lamm_kz(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_7792_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,D: fun_nat_complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(D),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_complex(C,I3) = aa_nat_complex(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_7793_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,D: fun_nat_real] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(D),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_real(C,I3) = aa_nat_real(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_7794_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups708209901874060359at_nat(aTP_Lamm_hy(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_7795_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups705719431365010083at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_7796_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B4: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_7797_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B4: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_7798_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B4: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_7799_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_7800_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(semiri681578069525770553at_rat,groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_7801_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_7802_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_7803_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_sg(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sh(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_7804_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_si(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sk(A),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_7805_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [I2: complex] :
            ( member_complex(I2,A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7806_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups129246275422532515t_real(F,A4)),groups129246275422532515t_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7807_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( finite_finite_int(A4)
     => ( ! [I2: int] :
            ( member_int(I2,A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7808_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A4)
     => ( ! [I2: real] :
            ( member_real(I2,A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_real )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7809_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [I2: complex] :
            ( member_complex(I2,A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7810_prod__mono__strict,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7811_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(A4)
     => ( ! [I2: int] :
            ( member_int(I2,A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7812_prod__mono__strict,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A4)
     => ( ! [I2: real] :
            ( member_real(I2,A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_real )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7813_prod__mono__strict,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( ! [I2: complex] :
            ( member_complex(I2,A4)
           => ( 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)) ) )
       => ( ( A4 != bot_bot_set_complex )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups861055069439313189ex_nat(F,A4)),groups861055069439313189ex_nat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7814_prod__mono__strict,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(A4)
     => ( ! [I2: int] :
            ( member_int(I2,A4)
           => ( 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,aa_int_nat(F,I2)),aa_int_nat(G,I2)) ) )
       => ( ( A4 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1707563613775114915nt_nat(F,A4)),groups1707563613775114915nt_nat(G,A4)) ) ) ) ).

% prod_mono_strict
tff(fact_7815_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups3455450783089532116nteger(F,A4))
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(F,X3)) ) ) ) ).

% even_prod_iff
tff(fact_7816_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_co8153928105989009288nteger] :
      ( finite3207457112153483333omplex(A4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),groups8682486955453173170nteger(F,A4))
      <=> ? [X3: complex] :
            ( member_complex(X3,A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_com4217951221296830281nteger(F,X3)) ) ) ) ).

% even_prod_iff
tff(fact_7817_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( 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,A4))
      <=> ? [X3: complex] :
            ( member_complex(X3,A4)
            & 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,X3)) ) ) ) ).

% even_prod_iff
tff(fact_7818_even__prod__iff,axiom,
    ! [A4: set_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(A4)
     => ( 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,A4))
      <=> ? [X3: complex] :
            ( member_complex(X3,A4)
            & 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,X3)) ) ) ) ).

% even_prod_iff
tff(fact_7819_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups708209901874060359at_nat(F,A4))
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & 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,X3)) ) ) ) ).

% even_prod_iff
tff(fact_7820_even__prod__iff,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( 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,A4))
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
            & 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,X3)) ) ) ) ).

% even_prod_iff
tff(fact_7821_even__prod__iff,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( finite_finite_int(A4)
     => ( 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,A4))
      <=> ? [X3: int] :
            ( member_int(X3,A4)
            & 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,X3)) ) ) ) ).

% even_prod_iff
tff(fact_7822_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_7823_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_7824_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_7825_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_7826_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_7827_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_7828_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_real,W: fun_int_real] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( member_int(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2316167850115554303t_real(Z,I6)),groups2316167850115554303t_real(W,I6)))),groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_sl(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7829_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_real,W: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1681761925125756287l_real(Z,I6)),groups1681761925125756287l_real(W,I6)))),groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_sm(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7830_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: fun_complex_real,W: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_complex_real(Z,I2))),one_one_real) )
     => ( ! [I2: complex] :
            ( member_complex(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_complex_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups766887009212190081x_real(Z,I6)),groups766887009212190081x_real(W,I6)))),groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_sn(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7831_norm__prod__diff,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,Z: fun_Pr672294278902479040t_real,W: fun_Pr672294278902479040t_real] :
      ( ! [I2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(I2,I6)
         => 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,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_Pro6168199777849092593t_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6036352826371341000t_real(Z,I6)),groups6036352826371341000t_real(W,I6)))),groups4567486121110086003t_real(aa_fun8712866722335801276t_real(aTP_Lamm_so(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7832_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_complex,W: fun_int_complex] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( member_int(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups7440179247065528705omplex(Z,I6)),groups7440179247065528705omplex(W,I6)))),groups8778361861064173332t_real(aa_fun1985759105522019690t_real(aTP_Lamm_sp(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7833_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_complex,W: fun_real_complex] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups713298508707869441omplex(Z,I6)),groups713298508707869441omplex(W,I6)))),groups8097168146408367636l_real(aa_fun2786211375485122666l_real(aTP_Lamm_sq(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7834_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: fun_complex_complex,W: fun_complex_complex] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Z,I2))),one_one_real) )
     => ( ! [I2: complex] :
            ( member_complex(I2,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups3708469109370488835omplex(Z,I6)),groups3708469109370488835omplex(W,I6)))),groups5808333547571424918x_real(aa_fun1953249243402654830x_real(aTP_Lamm_sr(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7835_norm__prod__diff,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,Z: fun_Pr4050773816208733250omplex,W: fun_Pr4050773816208733250omplex] :
      ( ! [I2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(I2,I6)
         => 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,I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_Pro82899191565270131omplex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups8110221916422527690omplex(Z,I6)),groups8110221916422527690omplex(W,I6)))),groups4567486121110086003t_real(aa_fun6819693891783611194t_real(aTP_Lamm_ss(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7836_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_real,W: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups129246275422532515t_real(Z,I6)),groups129246275422532515t_real(W,I6)))),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_st(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7837_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_complex,W: fun_nat_complex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I6)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(W,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups6464643781859351333omplex(Z,I6)),groups6464643781859351333omplex(W,I6)))),groups6591440286371151544t_real(aa_fun7054779078796324018t_real(aTP_Lamm_su(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_7838_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_complex(C,I3) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_7839_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
         => ( aa_nat_real(C,I3) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_7840_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,K2: nat] :
      ( ! [W2: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),W2),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_complex(C,K2) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_7841_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,K2: nat] :
      ( ! [W2: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),W2),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_real(C,K2) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_7842_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ii(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_7843_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ij(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_7844_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hy(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_7845_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_hz(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_7846_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_7847_sum__up__index__split,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_7848_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_7849_sum__up__index__split,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] : groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_7850_gbinomial__parallel__sum,axiom,
    ! [A: complex,Nb: nat] : groups2073611262835488442omplex(aTP_Lamm_sv(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),Nb) ).

% gbinomial_parallel_sum
tff(fact_7851_gbinomial__parallel__sum,axiom,
    ! [A: rat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_sw(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),Nb) ).

% gbinomial_parallel_sum
tff(fact_7852_gbinomial__parallel__sum,axiom,
    ! [A: real,Nb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sx,A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),Nb) ).

% gbinomial_parallel_sum
tff(fact_7853_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sy(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ta(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_7854_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sy(Nb)))) = groups6591440286371151544t_real(aTP_Lamm_tc(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_7855_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_7856_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_real] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( ! [B3: int] :
              ( member_int(B3,minus_minus_set_int(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,B3)) )
         => ( ! [A3: int] :
                ( member_int(A3,A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A4)),groups2316167850115554303t_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7857_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,B3)) )
         => ( ! [A3: real] :
                ( member_real(A3,A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A4)),groups1681761925125756287l_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7858_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( ! [B3: complex] :
              ( member_complex(B3,minus_811609699411566653omplex(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,B3)) )
         => ( ! [A3: complex] :
                ( member_complex(A3,A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A4)),groups766887009212190081x_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7859_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_rat] :
      ( finite_finite_int(B4)
     => ( ord_less_eq_set_int(A4,B4)
       => ( ! [B3: int] :
              ( member_int(B3,minus_minus_set_int(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B3)) )
         => ( ! [A3: int] :
                ( member_int(A3,A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A4)),groups1072433553688619179nt_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7860_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B3)) )
         => ( ! [A3: real] :
                ( member_real(A3,A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A4)),groups4061424788464935467al_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7861_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( ! [B3: complex] :
              ( member_complex(B3,minus_811609699411566653omplex(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B3)) )
         => ( ! [A3: complex] :
                ( member_complex(A3,A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A4)),groups225925009352817453ex_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7862_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A4,B4)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,B3)) )
         => ( ! [A3: real] :
                ( member_real(A3,A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A4)),groups4694064378042380927al_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7863_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(B4)
     => ( ord_le211207098394363844omplex(A4,B4)
       => ( ! [B3: complex] :
              ( member_complex(B3,minus_811609699411566653omplex(B4,A4))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,B3)) )
         => ( ! [A3: complex] :
                ( member_complex(A3,A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A3)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A4)),groups858564598930262913ex_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7864_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,B3)) )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A4)),groups129246275422532515t_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7865_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B3)) )
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A4)),groups73079841787564623at_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_7866_sum__gp__basic,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_7867_sum__gp__basic,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(one_one_rat,Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_7868_sum__gp__basic,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_7869_sum__gp__basic,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = 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,Nb))) ).

% sum_gp_basic
tff(fact_7870_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_td(C),Nb)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
          & ( aa_nat_complex(C,I3) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_7871_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_te(C),Nb)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
          & ( aa_nat_real(C,I3) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_7872_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K2: nat,Nb: nat] :
      ( ( aa_nat_complex(C,K2) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_td(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_7873_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K2: nat,Nb: nat] :
      ( ( aa_nat_real(C,K2) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_te(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_7874_pochhammer__Suc__prod,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_tf,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_7875_pochhammer__Suc__prod,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aTP_Lamm_tg(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_7876_pochhammer__Suc__prod,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_th,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_7877_pochhammer__Suc__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aTP_Lamm_ti(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_7878_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,Nb: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_7879_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,Nb: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Z4,A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_7880_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,Nb: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_7881_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,Nb: nat] :
      ( ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),A)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_7882_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,Nb: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z4: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_7883_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,Nb: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Z4,A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_7884_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,Nb: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_7885_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,Nb: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z4: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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,Z4),A)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(B3),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_7886_pochhammer__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_tl(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_7887_pochhammer__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_tm(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_7888_pochhammer__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tn(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_7889_pochhammer__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_to(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_7890_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_7891_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_7892_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_7893_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_7894_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_tp(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ta(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_7895_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_tp(Nb)))) = groups6591440286371151544t_real(aTP_Lamm_tc(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_7896_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)) ) ) ).

% fact_div_fact
tff(fact_7897_summable__Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_mg(A))
     => ( summable_real(aTP_Lamm_mg(B))
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tr(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_7898_summable__Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ts(A))
     => ( summable_real(aTP_Lamm_ts(B))
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_tu(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_7899_Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_mg(A))
     => ( summable_real(aTP_Lamm_mg(B))
       => ( aa_complex_complex(times_times_complex(suminf_complex(A)),suminf_complex(B)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tr(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_7900_Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ts(A))
     => ( summable_real(aTP_Lamm_ts(B))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(A)),suminf_real(B)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_tu(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_7901_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups129246275422532515t_real(aTP_Lamm_rg(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_7902_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_rh(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_7903_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups708209901874060359at_nat(aTP_Lamm_ri(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_7904_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_rj(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_7905_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_tv(F),A,B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_7906_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_fo1949268297981939178at_rat(aTP_Lamm_tw(F),A,B,zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7907_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_tx(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7908_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ty(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7909_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_tz(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_7910_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_je(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_7911_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_jf(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_7912_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_7913_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups6591440286371151544t_real(aTP_Lamm_jh(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_7914_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_complex,Nb: nat,B: fun_nat_complex,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_complex(A,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_ua(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_7915_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_rat,Nb: nat,B: fun_nat_rat,Xb: rat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_rat(A,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_uc(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_7916_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_int,Nb: nat,B: fun_nat_int,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_int(A,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ue(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_7917_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_real,Nb: nat,B: fun_nat_real,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_real(A,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_uf(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_7918_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_tl(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7919_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_tm(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7920_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tn(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7921_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_to(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_7922_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,Nb: nat,K2: complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K2
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K2 )
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))
           => ( aa_nat_complex(C,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_7923_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,Nb: nat,K2: real] :
      ( ! [X3: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),X3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K2
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K2 )
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))
           => ( aa_nat_real(C,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_7924_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_ug(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Mb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_7925_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_uh(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Mb)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_7926_gbinomial__sum__lower__neg,axiom,
    ! [A: real,Mb: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ui,A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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)),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_7927_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uj(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uj(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ul(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_7928_Cauchy__product__sums,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_mg(A))
     => ( summable_real(aTP_Lamm_mg(B))
       => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tr(A),B)),aa_complex_complex(times_times_complex(suminf_complex(A)),suminf_complex(B))) ) ) ).

% Cauchy_product_sums
tff(fact_7929_Cauchy__product__sums,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ts(A))
     => ( summable_real(aTP_Lamm_ts(B))
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_tu(A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(A)),suminf_real(B))) ) ) ).

% Cauchy_product_sums
tff(fact_7930_gbinomial__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K2)) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(aTP_Lamm_um(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7931_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_un(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7932_gbinomial__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_uo,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7933_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_up,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7934_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K2)) = aa_int_int(divide_divide_int(groups705719431365010083at_int(aTP_Lamm_uq(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K2))) ).

% gbinomial_Suc
tff(fact_7935_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ur(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rx(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7936_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_us(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sb(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7937_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ut(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sf(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7938_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uu(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sd(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7939_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_uv(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rz(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_7940_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: complex,Xb: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_uw(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ux(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_7941_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: rat,Xb: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_uy(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_uz(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_7942_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: real,Xb: real,Y: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_va(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_vb(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_7943_root__polyfun,axiom,
    ! [Nb: nat,Z: int,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_int(power_power_int(Z),Nb) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_vc(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_7944_root__polyfun,axiom,
    ! [Nb: nat,Z: complex,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_complex(power_power_complex(Z),Nb) = A )
      <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_vd(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_7945_root__polyfun,axiom,
    ! [Nb: nat,Z: rat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_rat(power_power_rat(Z),Nb) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ve(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_7946_root__polyfun,axiom,
    ! [Nb: nat,Z: code_integer,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),Nb) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_vf(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_7947_root__polyfun,axiom,
    ! [Nb: nat,Z: real,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) = A )
      <=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_vg(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_7948_sum__gp0,axiom,
    ! [Xb: complex,Nb: nat] :
      groups2073611262835488442omplex(power_power_complex(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp0
tff(fact_7949_sum__gp0,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),minus_minus_rat(one_one_rat,Xb))) ).

% sum_gp0
tff(fact_7950_sum__gp0,axiom,
    ! [Xb: real,Nb: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp0
tff(fact_7951_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups2073611262835488442omplex(aTP_Lamm_vh(Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_7952_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_vi(Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_7953_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vj,Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_7954_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: complex,Xb: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_uw(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_vk(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_7955_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: rat,Xb: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_uy(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_vl(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_7956_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: real,Xb: real,Y: real] : groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_va(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_vm(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_7957_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_vo(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_7958_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_rat,Xb: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xb,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_vq(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_7959_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_vs(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_7960_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_vu(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_7961_polyfun__extremal__lemma,axiom,
    ! [E: real,C: fun_nat_complex,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ? [M7: real] :
        ! [Z4: complex] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),aa_complex_real(real_V1022390504157884413omplex,Z4))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z4)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_7962_polyfun__extremal__lemma,axiom,
    ! [E: real,C: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ? [M7: real] :
        ! [Z4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),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,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Z4)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_7963_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_vw(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_7964_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_rat,Xb: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Xb,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_vy(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_7965_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_tk(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_wa(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_7966_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_wc(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_7967_fact__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7968_fact__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7969_fact__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7970_fact__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_7971_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Mb))),one_one_complex)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_7972_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Mb))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_7973_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : 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,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),one_one_real)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_7974_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_wd(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_7975_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_we,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_7976_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_wf(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_7977_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wg,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_7978_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_wh(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_7979_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_wi,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_7980_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_wj(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_7981_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wk,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_7982_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_7983_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_7984_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_7985_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_7986_binomial__n__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Nb) = one_one_nat ).

% binomial_n_n
tff(fact_7987_binomial__0__Suc,axiom,
    ! [K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K2)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_7988_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_7989_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% binomial_eq_0_iff
tff(fact_7990_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ).

% binomial_Suc_Suc
tff(fact_7991_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_7992_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_7993_prod__eq__1__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( ( groups861055069439313189ex_nat(F,A4) = one_one_nat )
      <=> ! [X3: complex] :
            ( member_complex(X3,A4)
           => ( aa_complex_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7994_prod__eq__1__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( groups708209901874060359at_nat(F,A4) = one_one_nat )
      <=> ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => ( aa_nat_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_7995_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% zero_less_binomial_iff
tff(fact_7996_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_7997_prod__pos__nat__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A4))
      <=> ! [X3: complex] :
            ( member_complex(X3,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7998_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A4))
      <=> ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_7999_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_8000_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_8001_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_8002_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_8003_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_8004_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_8005_binomial__eq__0,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8006_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))),aa_nat_nat(suc,K2)) ).

% Suc_times_binomial_eq
tff(fact_8007_Suc__times__binomial,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ).

% Suc_times_binomial
tff(fact_8008_binomial__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% binomial_symmetric
tff(fact_8009_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_8010_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_8011_zero__less__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ).

% zero_less_binomial
tff(fact_8012_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_8013_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))),aa_nat_nat(suc,K2)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8014_choose__mult,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K2))) ) ) ) ).

% choose_mult
tff(fact_8015_binomial__absorb__comp,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ).

% binomial_absorb_comp
tff(fact_8016_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_8017_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_8018_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_8019_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_8020_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_8021_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_8022_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_8023_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_8024_binomial__absorption,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ).

% binomial_absorption
tff(fact_8025_binomial__fact__lemma,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_8026_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ao,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_8027_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wm,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_8028_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wn,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_8029_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wn,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8030_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8031_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8032_binomial__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_8033_binomial__maximum_H,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_8034_binomial__maximum,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8035_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_antimono
tff(fact_8036_binomial__le__pow2,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_8037_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_8038_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_8039_choose__reduce__nat,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_8040_times__binomial__minus1__eq,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8041_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_8042_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_8043_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_8044_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_8045_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_8046_binomial__altdef__nat,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_8047_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wo(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_8048_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wp(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_8049_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ) ).

% binomial_less_binomial_Suc
tff(fact_8050_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_8051_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8052_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8053_binomial__addition__formula,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ).

% binomial_addition_formula
tff(fact_8054_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8055_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8056_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8057_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8058_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8059_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8060_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_8061_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_8062_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_8063_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_8064_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_8065_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sy(Nb)))) = groups708209901874060359at_nat(aTP_Lamm_wq(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_8066_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sy(Nb)))) = groups705719431365010083at_int(aTP_Lamm_ws(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_8067_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_8068_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wt(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_8069_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8070_binomial__ring,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),Nb) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wu(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8071_binomial__ring,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wv(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8072_binomial__ring,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ww(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8073_binomial__ring,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wt(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8074_binomial__ring,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Nb) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wx(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8075_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wy(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8076_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_wz(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8077_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Nb) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xa(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8078_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_8079_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_tp(Nb)))) = groups708209901874060359at_nat(aTP_Lamm_wq(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_8080_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_tp(Nb)))) = groups705719431365010083at_int(aTP_Lamm_ws(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_8081_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xb,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_8082_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_8083_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_xc(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8084_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_xd(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8085_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xe,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8086_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_xf(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8087_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xg,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8088_binomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% binomial_r_part_sum
tff(fact_8089_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xh,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_8090_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_8091_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8092_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_8093_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_8094_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_8095_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),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),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_8096_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups2073611262835488442omplex(aTP_Lamm_xi(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_8097_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups7501900531339628137nteger(aTP_Lamm_xj(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8098_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_xk,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8099_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_xl(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_8100_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xm,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8101_binomial__code,axiom,
    ! [Nb: nat,K2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K2))) ) ).

% binomial_code
tff(fact_8102_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_8103_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_8104_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)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,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_8105_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)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,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_8106_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_8107_sin__x__sin__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_xo(Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_x_sin_y
tff(fact_8108_sin__x__sin__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_xq(Xb),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Y))) ).

% sin_x_sin_y
tff(fact_8109_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nu,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_8110_sums__cos__x__plus__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_xs(Xb),Y)),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))) ).

% sums_cos_x_plus_y
tff(fact_8111_sums__cos__x__plus__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_xu(Xb),Y)),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y))) ).

% sums_cos_x_plus_y
tff(fact_8112_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(bitM(Mb),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_8113_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_8114_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_8115_mult__scaleR__right,axiom,
    ! [Xb: real,A: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) ).

% mult_scaleR_right
tff(fact_8116_mult__scaleR__right,axiom,
    ! [Xb: complex,A: real,Y: complex] : aa_complex_complex(times_times_complex(Xb),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Y)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(times_times_complex(Xb),Y)) ).

% mult_scaleR_right
tff(fact_8117_mult__scaleR__left,axiom,
    ! [A: real,Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) ).

% mult_scaleR_left
tff(fact_8118_mult__scaleR__left,axiom,
    ! [A: real,Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Xb)),Y) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(times_times_complex(Xb),Y)) ).

% mult_scaleR_left
tff(fact_8119_inverse__mult__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_8120_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% inverse_mult_distrib
tff(fact_8121_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).

% inverse_mult_distrib
tff(fact_8122_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_8123_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_8124_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_8125_inverse__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(inverse_inverse_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_8126_inverse__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_8127_inverse__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,Xb) = one_one_rat )
    <=> ( Xb = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_8128_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) ).

% inverse_divide
tff(fact_8129_inverse__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(B),A) ).

% inverse_divide
tff(fact_8130_inverse__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,divide_divide_rat(A,B)) = divide_divide_rat(B,A) ).

% inverse_divide
tff(fact_8131_scaleR__one,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,one_one_real),Xb) = Xb ).

% scaleR_one
tff(fact_8132_scaleR__one,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,one_one_real),Xb) = Xb ).

% scaleR_one
tff(fact_8133_scaleR__scaleR,axiom,
    ! [A: real,B: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Xb) ).

% scaleR_scaleR
tff(fact_8134_scaleR__scaleR,axiom,
    ! [A: real,B: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B),Xb)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Xb) ).

% scaleR_scaleR
tff(fact_8135_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8136_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8137_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8138_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8139_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_8140_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_8141_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8142_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8143_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% inverse_negative_iff_negative
tff(fact_8144_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% inverse_negative_iff_negative
tff(fact_8145_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% inverse_positive_iff_positive
tff(fact_8146_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% inverse_positive_iff_positive
tff(fact_8147_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_8148_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_8149_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_8150_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8151_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8152_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8153_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8154_of__int__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K2)) = numera6690914467698888265omplex(K2) ).

% of_int_numeral
tff(fact_8155_of__int__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K2)) = numeral_numeral_real(K2) ).

% of_int_numeral
tff(fact_8156_of__int__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K2)) = numeral_numeral_rat(K2) ).

% of_int_numeral
tff(fact_8157_of__int__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,K2) ).

% of_int_numeral
tff(fact_8158_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),B)) )
    <=> ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
tff(fact_8159_scaleR__eq__iff,axiom,
    ! [B: complex,U: real,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,U),A)) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,U),B)) )
    <=> ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
tff(fact_8160_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_8161_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_8162_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_8163_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_8164_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_8165_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_8166_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_8167_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_8168_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_8169_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_8170_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_8171_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_8172_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_8173_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_8174_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_8175_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_8176_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_8177_scaleR__power,axiom,
    ! [Xb: real,Y: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Xb),Y)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ).

% scaleR_power
tff(fact_8178_scaleR__power,axiom,
    ! [Xb: real,Y: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,Xb),Y)),Nb) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) ).

% scaleR_power
tff(fact_8179_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_8180_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_8181_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_8182_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Nb) ).

% of_int_power
tff(fact_8183_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_8184_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_8185_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_8186_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8187_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8188_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8189_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8190_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xb) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8191_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8192_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8193_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8194_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(bitM(K2)) ).

% dbl_dec_simps(5)
tff(fact_8195_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K2)) = numeral_numeral_real(bitM(K2)) ).

% dbl_dec_simps(5)
tff(fact_8196_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K2)) = numeral_numeral_rat(bitM(K2)) ).

% dbl_dec_simps(5)
tff(fact_8197_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,bitM(K2)) ).

% dbl_dec_simps(5)
tff(fact_8198_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_8199_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_8200_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8201_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8202_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_8203_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_8204_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_8205_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_8206_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_8207_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_8208_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(W)) ).

% inverse_eq_divide_numeral
tff(fact_8209_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W)) ).

% inverse_eq_divide_numeral
tff(fact_8210_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,numeral_numeral_rat(W)) = divide_divide_rat(one_one_rat,numeral_numeral_rat(W)) ).

% inverse_eq_divide_numeral
tff(fact_8211_scaleR__minus1__left,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb) = aa_real_real(uminus_uminus_real,Xb) ).

% scaleR_minus1_left
tff(fact_8212_scaleR__minus1__left,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(uminus_uminus_real,one_one_real)),Xb) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% scaleR_minus1_left
tff(fact_8213_scaleR__collapse,axiom,
    ! [U: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),U)),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A)) = A ).

% scaleR_collapse
tff(fact_8214_scaleR__collapse,axiom,
    ! [U: real,A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),U)),A)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,U),A)) = A ).

% scaleR_collapse
tff(fact_8215_norm__scaleR,axiom,
    ! [A: real,Xb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(real_V7735802525324610683m_real,Xb)) ).

% norm_scaleR
tff(fact_8216_norm__scaleR,axiom,
    ! [A: real,Xb: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% norm_scaleR
tff(fact_8217_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K2)) = aa_num_nat(numeral_numeral_nat,bitM(K2)) ).

% pred_numeral_simps(2)
tff(fact_8218_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8219_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8220_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,uminus_uminus_rat(numeral_numeral_rat(W))) = divide_divide_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8221_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_8222_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_8223_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_8224_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_8225_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_8226_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_8227_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_8228_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_8229_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_8230_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_8231_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_8232_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_8233_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_8234_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_8235_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_8236_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_8237_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_8238_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_8239_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_8240_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_8241_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_8242_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_8243_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_8244_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_8245_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_8246_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_8247_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_8248_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_8249_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_8250_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_8251_scaleR__times,axiom,
    ! [U: num,W: num,A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,numeral_numeral_real(U)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),A) ).

% scaleR_times
tff(fact_8252_scaleR__times,axiom,
    ! [U: num,W: num,A: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,numeral_numeral_real(U)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),A) ).

% scaleR_times
tff(fact_8253_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_8254_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_8255_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_8256_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_8257_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_8258_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_8259_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_8260_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_8261_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_8262_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_8263_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_8264_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_8265_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8266_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8267_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8268_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8269_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8270_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8271_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8272_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8273_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_8274_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_8275_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_8276_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_8277_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_8278_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_8279_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_8280_tan__periodic__int,axiom,
    ! [Xb: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_int
tff(fact_8281_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(W)),numeral_numeral_real(V))),A) ).

% inverse_scaleR_times
tff(fact_8282_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(W)),numeral_numeral_real(V))),A) ).

% inverse_scaleR_times
tff(fact_8283_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(U)),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),numeral_numeral_real(V))),A) ).

% fraction_scaleR_times
tff(fact_8284_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(U)),numeral_numeral_real(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(U)),numeral_numeral_real(W))),numeral_numeral_real(V))),A) ).

% fraction_scaleR_times
tff(fact_8285_scaleR__half__double,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)) = A ).

% scaleR_half_double
tff(fact_8286_scaleR__half__double,axiom,
    ! [A: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_complex(plus_plus_complex(A),A)) = A ).

% scaleR_half_double
tff(fact_8287_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_8288_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_8289_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_8290_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_8291_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_8292_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_8293_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_8294_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_8295_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_8296_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_8297_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_8298_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_8299_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8300_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8301_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8302_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8303_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8304_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8305_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8306_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8307_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8308_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8309_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8310_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_8311_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8312_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8313_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8314_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8315_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8316_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8317_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8318_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8319_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8320_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8321_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8322_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8323_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8324_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8325_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8326_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),aa_int_Code_integer(ring_18347121197199848620nteger,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8327_real__scaleR__def,axiom,
    ! [A: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Xb) ).

% real_scaleR_def
tff(fact_8328_mult__of__int__commute,axiom,
    ! [Xb: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,Xb)) ).

% mult_of_int_commute
tff(fact_8329_mult__of__int__commute,axiom,
    ! [Xb: int,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,Xb)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_int_rat(ring_1_of_int_rat,Xb)) ).

% mult_of_int_commute
tff(fact_8330_mult__of__int__commute,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,Xb)) ).

% mult_of_int_commute
tff(fact_8331_power__inverse,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,A)),Nb) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_inverse
tff(fact_8332_power__inverse,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,A)),Nb) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(power_power_complex(A),Nb)) ).

% power_inverse
tff(fact_8333_power__inverse,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,A)),Nb) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% power_inverse
tff(fact_8334_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_8335_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_8336_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_8337_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_8338_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_8339_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_8340_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8341_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8342_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8343_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Y)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8344_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,Xb: complex] :
      ( ( aa_complex_complex(times_times_complex(Y),Xb) = aa_complex_complex(times_times_complex(Xb),Y) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y)),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8345_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,Xb: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,Y)),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(inverse_inverse_rat,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8346_scaleR__right__diff__distrib,axiom,
    ! [A: real,Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) ).

% scaleR_right_diff_distrib
tff(fact_8347_scaleR__right__diff__distrib,axiom,
    ! [A: real,Xb: complex,Y: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Xb)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Y)) ).

% scaleR_right_diff_distrib
tff(fact_8348_scaleR__right__distrib,axiom,
    ! [A: real,Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) ).

% scaleR_right_distrib
tff(fact_8349_scaleR__right__distrib,axiom,
    ! [A: real,Xb: complex,Y: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Xb)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Y)) ).

% scaleR_right_distrib
tff(fact_8350_semiring__norm_I26_J,axiom,
    bitM(one) = one ).

% semiring_norm(26)
tff(fact_8351_real__vector__affinity__eq,axiom,
    ! [Mb: real,Xb: real,C: real,Y: real] :
      ( ( Mb != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Mb),Xb)),C) = Y )
      <=> ( Xb = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,Mb)),Y)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,Mb)),C)) ) ) ) ).

% real_vector_affinity_eq
tff(fact_8352_real__vector__affinity__eq,axiom,
    ! [Mb: real,Xb: complex,C: complex,Y: complex] :
      ( ( Mb != zero_zero_real )
     => ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,Mb),Xb)),C) = Y )
      <=> ( Xb = aa_complex_complex(minus_minus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,Mb)),Y)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,Mb)),C)) ) ) ) ).

% real_vector_affinity_eq
tff(fact_8353_real__vector__eq__affinity,axiom,
    ! [Mb: real,Y: real,Xb: real,C: real] :
      ( ( Mb != zero_zero_real )
     => ( ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Mb),Xb)),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,Mb)),Y)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,Mb)),C)) = Xb ) ) ) ).

% real_vector_eq_affinity
tff(fact_8354_real__vector__eq__affinity,axiom,
    ! [Mb: real,Y: complex,Xb: complex,C: complex] :
      ( ( Mb != zero_zero_real )
     => ( ( Y = aa_complex_complex(plus_plus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,Mb),Xb)),C) )
      <=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,Mb)),Y)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,Mb)),C)) = Xb ) ) ) ).

% real_vector_eq_affinity
tff(fact_8355_pos__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_divideR_le_eq
tff(fact_8356_pos__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% pos_le_divideR_eq
tff(fact_8357_neg__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% neg_divideR_le_eq
tff(fact_8358_neg__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_le_divideR_eq
tff(fact_8359_neg__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_less_divideR_eq
tff(fact_8360_neg__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% neg_divideR_less_eq
tff(fact_8361_pos__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),B) ) ) ).

% pos_less_divideR_eq
tff(fact_8362_pos__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_divideR_less_eq
tff(fact_8363_summable__exp__generic,axiom,
    ! [Xb: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_xv,Xb)) ).

% summable_exp_generic
tff(fact_8364_summable__exp__generic,axiom,
    ! [Xb: complex] : summable_complex(aTP_Lamm_xw(Xb)) ).

% summable_exp_generic
tff(fact_8365_neg__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_8366_neg__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_8367_pos__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_8368_pos__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_8369_pos__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_8370_pos__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_8371_neg__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_8372_neg__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_8373_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_8374_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_8375_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less
tff(fact_8376_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less
tff(fact_8377_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8378_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8379_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8380_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8381_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% inverse_negative_imp_negative
tff(fact_8382_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% inverse_negative_imp_negative
tff(fact_8383_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_8384_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_8385_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).

% negative_imp_inverse_negative
tff(fact_8386_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).

% negative_imp_inverse_negative
tff(fact_8387_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).

% positive_imp_inverse_positive
tff(fact_8388_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).

% positive_imp_inverse_positive
tff(fact_8389_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8390_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8391_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8392_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,numeral_numeral_real(one)) = numeral_numeral_real(one) ).

% inverse_numeral_1
tff(fact_8393_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_8394_inverse__numeral__1,axiom,
    aa_rat_rat(inverse_inverse_rat,numeral_numeral_rat(one)) = numeral_numeral_rat(one) ).

% inverse_numeral_1
tff(fact_8395_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B ) ) ).

% inverse_unique
tff(fact_8396_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = one_one_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = B ) ) ).

% inverse_unique
tff(fact_8397_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = one_one_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).

% inverse_unique
tff(fact_8398_divide__inverse__commute,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B)),A) ).

% divide_inverse_commute
tff(fact_8399_divide__inverse__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),A) ).

% divide_inverse_commute
tff(fact_8400_divide__inverse__commute,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B)),A) ).

% divide_inverse_commute
tff(fact_8401_divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% divide_inverse
tff(fact_8402_divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% divide_inverse
tff(fact_8403_divide__inverse,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,B)) ).

% divide_inverse
tff(fact_8404_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B)) ).

% field_class.field_divide_inverse
tff(fact_8405_field__class_Ofield__divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% field_class.field_divide_inverse
tff(fact_8406_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B: rat] : divide_divide_rat(A,B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,B)) ).

% field_class.field_divide_inverse
tff(fact_8407_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ).

% inverse_eq_divide
tff(fact_8408_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_8409_inverse__eq__divide,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ).

% inverse_eq_divide
tff(fact_8410_power__mult__power__inverse__commute,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8411_power__mult__power__inverse__commute,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8412_power__mult__power__inverse__commute,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8413_power__mult__inverse__distrib,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8414_power__mult__inverse__distrib,axiom,
    ! [Xb: complex,Mb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8415_power__mult__inverse__distrib,axiom,
    ! [Xb: rat,Mb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(inverse_inverse_rat,Xb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,Xb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8416_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_8417_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_8418_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_8419_scaleR__left__distrib,axiom,
    ! [A: real,B: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Xb)) ).

% scaleR_left_distrib
tff(fact_8420_scaleR__left__distrib,axiom,
    ! [A: real,B: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Xb) = aa_complex_complex(plus_plus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Xb)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B),Xb)) ).

% scaleR_left_distrib
tff(fact_8421_scaleR__left_Oadd,axiom,
    ! [Xb: real,Y: real,Xaa: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),Xaa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Xb),Xaa)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Y),Xaa)) ).

% scaleR_left.add
tff(fact_8422_scaleR__left_Oadd,axiom,
    ! [Xb: real,Y: real,Xaa: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),Xaa) = aa_complex_complex(plus_plus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,Xb),Xaa)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,Y),Xaa)) ).

% scaleR_left.add
tff(fact_8423_divide__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_8424_scaleR__left__diff__distrib,axiom,
    ! [A: real,B: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Xb)) ).

% scaleR_left_diff_distrib
tff(fact_8425_scaleR__left__diff__distrib,axiom,
    ! [A: real,B: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),Xb) = aa_complex_complex(minus_minus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Xb)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B),Xb)) ).

% scaleR_left_diff_distrib
tff(fact_8426_scaleR__left_Odiff,axiom,
    ! [Xb: real,Y: real,Xaa: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),Xaa) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Xb),Xaa)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Y),Xaa)) ).

% scaleR_left.diff
tff(fact_8427_scaleR__left_Odiff,axiom,
    ! [Xb: real,Y: real,Xaa: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),Xaa) = aa_complex_complex(minus_minus_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,Xb),Xaa)),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,Y),Xaa)) ).

% scaleR_left.diff
tff(fact_8428_exp__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_xv,Xb)),aa_real_real(exp_real,Xb)) ).

% exp_converges
tff(fact_8429_exp__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_xw(Xb)),aa_complex_complex(exp_complex,Xb)) ).

% exp_converges
tff(fact_8430_exp__def,axiom,
    ! [X: real] : aa_real_real(exp_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xv,X)) ).

% exp_def
tff(fact_8431_exp__def,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) = suminf_complex(aTP_Lamm_xw(X)) ).

% exp_def
tff(fact_8432_summable__norm__exp,axiom,
    ! [Xb: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_xx,Xb)) ).

% summable_norm_exp
tff(fact_8433_summable__norm__exp,axiom,
    ! [Xb: complex] : summable_real(aTP_Lamm_xy(Xb)) ).

% summable_norm_exp
tff(fact_8434_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),B)) ).

% complex_scaleR
tff(fact_8435_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(aa_num_num(bit0,Nb)) = aa_num_num(bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_8436_semiring__norm_I28_J,axiom,
    ! [Nb: num] : bitM(aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_8437_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_8438_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_8439_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le
tff(fact_8440_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le
tff(fact_8441_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_8442_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_8443_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_8444_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_8445_inverse__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_8446_inverse__le__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_8447_one__less__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_less_inverse
tff(fact_8448_one__less__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_less_inverse
tff(fact_8449_one__less__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_8450_one__less__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_8451_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_8452_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_8453_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_8454_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8455_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8456_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8457_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_8458_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% inverse_add
tff(fact_8459_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_8460_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8461_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B),A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8462_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),minus_minus_rat(B,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8463_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8464_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8465_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = divide_divide_rat(one_one_rat,A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8466_scaleR__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),C)) ) ) ).

% scaleR_right_mono_neg
tff(fact_8467_scaleR__right__mono,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Xb)) ) ) ).

% scaleR_right_mono
tff(fact_8468_scaleR__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_8469_scaleR__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_8470_scaleR__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_8471_scaleR__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,C),B)) ) ) ).

% scaleR_left_mono_neg
tff(fact_8472_scaleR__left__mono,axiom,
    ! [Xb: real,Y: real,A: 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,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) ) ) ).

% scaleR_left_mono
tff(fact_8473_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_8474_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_8475_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_8476_real__of__int__div,axiom,
    ! [D: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Nb)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_8477_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bitM(Nb))) ).

% eval_nat_numeral(2)
tff(fact_8478_exp__series__add__commuting,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),Nb)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xz(Xb),Y),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ) ) ).

% exp_series_add_commuting
tff(fact_8479_exp__series__add__commuting,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
     => ( aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),Y)),Nb)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ya(Xb),Y),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ) ) ).

% exp_series_add_commuting
tff(fact_8480_exp__first__term,axiom,
    ! [Xb: real] : aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),suminf_real(aa_real_fun_nat_real(aTP_Lamm_yb,Xb))) ).

% exp_first_term
tff(fact_8481_exp__first__term,axiom,
    ! [Xb: complex] : aa_complex_complex(exp_complex,Xb) = aa_complex_complex(plus_plus_complex(one_one_complex),suminf_complex(aTP_Lamm_yc(Xb))) ).

% exp_first_term
tff(fact_8482_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_8483_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_8484_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_8485_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_8486_one__le__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_le_inverse
tff(fact_8487_one__le__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_le_inverse
tff(fact_8488_inverse__less__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_8489_inverse__less__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_8490_one__le__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_8491_one__le__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ) ).

% one_le_inverse_iff
tff(fact_8492_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),bitM(Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_8493_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bitM(Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_8494_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8495_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8496_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A),aa_rat_rat(inverse_inverse_rat,B)) = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),minus_minus_rat(A,B))),aa_rat_rat(inverse_inverse_rat,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8497_reals__Archimedean,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),Xb) ) ).

% reals_Archimedean
tff(fact_8498_reals__Archimedean,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),Xb) ) ).

% reals_Archimedean
tff(fact_8499_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_8500_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_8501_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_8502_of__int__leD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_leD
tff(fact_8503_of__int__leD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_8504_of__int__leD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% of_int_leD
tff(fact_8505_of__int__leD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_8506_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_8507_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_8508_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_8509_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
tff(fact_8510_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
tff(fact_8511_of__int__lessD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_lessD
tff(fact_8512_of__int__lessD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_8513_of__int__lessD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% of_int_lessD
tff(fact_8514_of__int__lessD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_8515_scaleR__mono,axiom,
    ! [A: real,B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Y)) ) ) ) ) ).

% scaleR_mono
tff(fact_8516_scaleR__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),D)) ) ) ) ) ).

% scaleR_mono'
tff(fact_8517_split__scaleR__neg__le,axiom,
    ! [A: real,Xb: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),zero_zero_real) ) ).

% split_scaleR_neg_le
tff(fact_8518_split__scaleR__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)) ) ).

% split_scaleR_pos_le
tff(fact_8519_scaleR__nonneg__nonneg,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_8520_scaleR__nonneg__nonpos,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),zero_zero_real) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_8521_scaleR__nonpos__nonneg,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),zero_zero_real) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_8522_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_8523_scaleR__left__le__one__le,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),Xb) ) ) ).

% scaleR_left_le_one_le
tff(fact_8524_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_8525_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_8526_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X4: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),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,X4),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 = X4 ) ) ) ).

% floor_exists1
tff(fact_8527_floor__exists1,axiom,
    ! [Xb: rat] :
    ? [X4: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X4)),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,X4),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 = X4 ) ) ) ).

% floor_exists1
tff(fact_8528_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K2)) ).

% of_int_neg_numeral
tff(fact_8529_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)) ).

% of_int_neg_numeral
tff(fact_8530_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2)) ).

% of_int_neg_numeral
tff(fact_8531_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = uminus_uminus_rat(numeral_numeral_rat(K2)) ).

% of_int_neg_numeral
tff(fact_8532_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2)) ).

% of_int_neg_numeral
tff(fact_8533_scaleR__2,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb) ).

% scaleR_2
tff(fact_8534_scaleR__2,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,numeral_numeral_real(aa_num_num(bit0,one))),Xb) = aa_complex_complex(plus_plus_complex(Xb),Xb) ).

% scaleR_2
tff(fact_8535_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_8536_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_8537_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_8538_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [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),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_8539_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
    <=> ? [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))),E) ) ) ).

% real_arch_inverse
tff(fact_8540_forall__pos__mono,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [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),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono
tff(fact_8541_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).

% int_le_real_less
tff(fact_8542_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).

% int_less_real_le
tff(fact_8543_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I3: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),pi) ) ).

% sin_zero_iff_int2
tff(fact_8544_exp__first__terms,axiom,
    ! [Xb: real,K2: nat] : aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xv,Xb),aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_real(aa_nat_fun_nat_real(aTP_Lamm_yd(Xb),K2))) ).

% exp_first_terms
tff(fact_8545_exp__first__terms,axiom,
    ! [Xb: complex,K2: nat] : aa_complex_complex(exp_complex,Xb) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(aTP_Lamm_xw(Xb),aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ye(Xb),K2))) ).

% exp_first_terms
tff(fact_8546_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(Xb),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_8547_summable__exp,axiom,
    ! [Xb: complex] : summable_complex(aTP_Lamm_yf(Xb)) ).

% summable_exp
tff(fact_8548_summable__exp,axiom,
    ! [Xb: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_yg,Xb)) ).

% summable_exp
tff(fact_8549_numeral__BitM,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(bitM(Nb)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,Nb))),one_one_complex) ).

% numeral_BitM
tff(fact_8550_numeral__BitM,axiom,
    ! [Nb: num] : numeral_numeral_real(bitM(Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(aa_num_num(bit0,Nb))),one_one_real) ).

% numeral_BitM
tff(fact_8551_numeral__BitM,axiom,
    ! [Nb: num] : numeral_numeral_rat(bitM(Nb)) = minus_minus_rat(numeral_numeral_rat(aa_num_num(bit0,Nb)),one_one_rat) ).

% numeral_BitM
tff(fact_8552_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,bitM(Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),one_one_int) ).

% numeral_BitM
tff(fact_8553_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(bitM(W))) ).

% odd_numeral_BitM
tff(fact_8554_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,bitM(W))) ).

% odd_numeral_BitM
tff(fact_8555_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,bitM(W))) ).

% odd_numeral_BitM
tff(fact_8556_ex__inverse__of__nat__less,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_8557_ex__inverse__of__nat__less,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_8558_power__diff__conv__inverse,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8559_power__diff__conv__inverse,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_complex(power_power_complex(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8560_power__diff__conv__inverse,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_rat(power_power_rat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8561_sin__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_yh,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_converges
tff(fact_8562_sin__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_yi(Xb)),aa_complex_complex(sin_complex,Xb)) ).

% sin_converges
tff(fact_8563_sin__def,axiom,
    ! [X: real] : aa_real_real(sin_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_yh,X)) ).

% sin_def
tff(fact_8564_sin__def,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,X) = suminf_complex(aTP_Lamm_yi(X)) ).

% sin_def
tff(fact_8565_cos__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_yj,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_converges
tff(fact_8566_cos__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_yk(Xb)),aa_complex_complex(cos_complex,Xb)) ).

% cos_converges
tff(fact_8567_cos__def,axiom,
    ! [X: real] : aa_real_real(cos_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_yj,X)) ).

% cos_def
tff(fact_8568_cos__def,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,X) = suminf_complex(aTP_Lamm_yk(X)) ).

% cos_def
tff(fact_8569_summable__norm__sin,axiom,
    ! [Xb: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_yl,Xb)) ).

% summable_norm_sin
tff(fact_8570_summable__norm__sin,axiom,
    ! [Xb: complex] : summable_real(aTP_Lamm_ym(Xb)) ).

% summable_norm_sin
tff(fact_8571_summable__norm__cos,axiom,
    ! [Xb: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_yn,Xb)) ).

% summable_norm_cos
tff(fact_8572_summable__norm__cos,axiom,
    ! [Xb: complex] : summable_real(aTP_Lamm_yo(Xb)) ).

% summable_norm_cos
tff(fact_8573_exp__first__two__terms,axiom,
    ! [Xb: 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)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_yp,Xb))) ).

% exp_first_two_terms
tff(fact_8574_exp__first__two__terms,axiom,
    ! [Xb: complex] : aa_complex_complex(exp_complex,Xb) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),Xb)),suminf_complex(aTP_Lamm_yq(Xb))) ).

% exp_first_two_terms
tff(fact_8575_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xb)))) ).

% real_of_int_div2
tff(fact_8576_real__of__int__div3,axiom,
    ! [Nb: 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,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(Nb),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_8577_sin__minus__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_yr,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_minus_converges
tff(fact_8578_sin__minus__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_ys(Xb)),aa_complex_complex(sin_complex,Xb)) ).

% sin_minus_converges
tff(fact_8579_cos__minus__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_yt,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_minus_converges
tff(fact_8580_cos__minus__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_yu(Xb)),aa_complex_complex(cos_complex,Xb)) ).

% cos_minus_converges
tff(fact_8581_even__of__int__iff,axiom,
    ! [K2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_of_int_iff
tff(fact_8582_even__of__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_of_int_iff
tff(fact_8583_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_8584_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,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_8585_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_8586_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ? [X3: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X3)),numeral_numeral_real(aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_8587_tan__cot,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xb)) ).

% tan_cot
tff(fact_8588_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)))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_8589_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)))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_8590_tan__sec,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( 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)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(cos_real,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_8591_tan__sec,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_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)))) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_8592_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_8593_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I3: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_8594_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8595_cos__x__cos__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_yw(Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ).

% cos_x_cos_y
tff(fact_8596_cos__x__cos__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_yy(Xb),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ).

% cos_x_cos_y
tff(fact_8597_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_real(ring_1_of_int_real,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K2),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_8598_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_int(ring_1_of_int_int,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K2))),
          $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(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K2),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_8599_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_complex(ring_17405671764205052669omplex,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K2),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_8600_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_rat(ring_1_of_int_rat,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: rat,
            l:= aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_8601_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_Code_integer(ring_18347121197199848620nteger,K2) = $ite(
        K2 = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_8602_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),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),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Y ) ) ) ).

% round_unique
tff(fact_8603_round__unique,axiom,
    ! [Xb: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
       => ( archim7778729529865785530nd_rat(Xb) = Y ) ) ) ).

% round_unique
tff(fact_8604_round__unique_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xb) = Nb ) ) ).

% round_unique'
tff(fact_8605_round__unique_H,axiom,
    ! [Xb: rat,Nb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xb,aa_int_rat(ring_1_of_int_rat,Nb)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))
     => ( archim7778729529865785530nd_rat(Xb) = Nb ) ) ).

% round_unique'
tff(fact_8606_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),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_8607_of__int__round__abs__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb)),Xb))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_8608_sinh__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_yz,Xb)),aa_real_real(sinh_real,Xb)) ).

% sinh_converges
tff(fact_8609_sinh__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_za(Xb)),aa_complex_complex(sinh_complex,Xb)) ).

% sinh_converges
tff(fact_8610_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_8611_round__numeral,axiom,
    ! [Nb: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_8612_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_8613_round__1,axiom,
    archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).

% round_1
tff(fact_8614_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_8615_round__neg__numeral,axiom,
    ! [Nb: num] : archim7778729529865785530nd_rat(uminus_uminus_rat(numeral_numeral_rat(Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_8616_divide__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_8617_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,archim7778729529865785530nd_rat(Xb)),archim7778729529865785530nd_rat(Y)) ) ).

% round_mono
tff(fact_8618_round__diff__minimal,axiom,
    ! [Z: real,Mb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(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,Mb)))) ).

% round_diff_minimal
tff(fact_8619_round__diff__minimal,axiom,
    ! [Z: rat,Mb: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Z))))),abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,Mb)))) ).

% round_diff_minimal
tff(fact_8620_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_8621_sinh__field__def,axiom,
    ! [Z: real] : aa_real_real(sinh_real,Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Z)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_8622_sinh__field__def,axiom,
    ! [Z: complex] : aa_complex_complex(sinh_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Z)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_8623_sinh__def,axiom,
    ! [Xb: real] : aa_real_real(sinh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% sinh_def
tff(fact_8624_sinh__def,axiom,
    ! [Xb: complex] : aa_complex_complex(sinh_complex,Xb) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).

% sinh_def
tff(fact_8625_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))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8626_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),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_8627_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,archim7778729529865785530nd_rat(Xb))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_8628_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),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_8629_of__int__round__ge,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))) ).

% of_int_round_ge
tff(fact_8630_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),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_8631_of__int__round__gt,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))) ).

% of_int_round_gt
tff(fact_8632_cosh__converges,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_zb,Xb)),aa_real_real(cosh_real,Xb)) ).

% cosh_converges
tff(fact_8633_cosh__converges,axiom,
    ! [Xb: complex] : aa_complex_o(sums_complex(aTP_Lamm_zc(Xb)),aa_complex_complex(cosh_complex,Xb)) ).

% cosh_converges
tff(fact_8634_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)),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_8635_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_8636_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,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,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_8637_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_8638_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_8639_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_8640_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_8641_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_8642_complex__i__mult__minus,axiom,
    ! [Xb: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),Xb)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% complex_i_mult_minus
tff(fact_8643_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_8644_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_8645_one__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_8646_log__less__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_8647_log__less__cancel__iff,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),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,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_8648_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_8649_divide__i,axiom,
    ! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xb) ).

% divide_i
tff(fact_8650_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_8651_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_8652_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_8653_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_8654_one__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_8655_log__le__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_8656_log__le__cancel__iff,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),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,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_8657_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_8658_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_8659_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_8660_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_8661_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_8662_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_8663_log__ln,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(exp_real,one_one_real)),Xb) ).

% log_ln
tff(fact_8664_cosh__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% cosh_add
tff(fact_8665_sinh__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% sinh_add
tff(fact_8666_sinh__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% sinh_diff
tff(fact_8667_cosh__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% cosh_diff
tff(fact_8668_Complex__eq__i,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_complex(aa_rea2649881322057434177omplex(complex2,Xb),Y) = imaginary_unit )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_8669_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(aa_rea2649881322057434177omplex(complex2,zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_8670_sinh__plus__cosh,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(cosh_complex,Xb)) = aa_complex_complex(exp_complex,Xb) ).

% sinh_plus_cosh
tff(fact_8671_sinh__plus__cosh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) = aa_real_real(exp_real,Xb) ).

% sinh_plus_cosh
tff(fact_8672_cosh__plus__sinh,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cosh_complex,Xb)),aa_complex_complex(sinh_complex,Xb)) = aa_complex_complex(exp_complex,Xb) ).

% cosh_plus_sinh
tff(fact_8673_cosh__plus__sinh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Xb)) = aa_real_real(exp_real,Xb) ).

% cosh_plus_sinh
tff(fact_8674_tanh__def,axiom,
    ! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(cosh_complex,Xb)) ).

% tanh_def
tff(fact_8675_tanh__def,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% tanh_def
tff(fact_8676_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)),imaginary_unit) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_8677_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_8678_log__base__change,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_8679_log__of__power__eq,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_8680_less__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).

% less_log_of_power
tff(fact_8681_cosh__minus__sinh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Xb)) = aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)) ).

% cosh_minus_sinh
tff(fact_8682_cosh__minus__sinh,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cosh_complex,Xb)),aa_complex_complex(sinh_complex,Xb)) = aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)) ).

% cosh_minus_sinh
tff(fact_8683_sinh__minus__cosh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ).

% sinh_minus_cosh
tff(fact_8684_sinh__minus__cosh,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(cosh_complex,Xb)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb))) ).

% sinh_minus_cosh
tff(fact_8685_log__mult,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_8686_log__divide,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_8687_le__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).

% le_log_of_power
tff(fact_8688_log__nat__power,axiom,
    ! [Xb: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_nat_power
tff(fact_8689_log__inverse,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_8690_cot__def,axiom,
    ! [X: complex] : aa_complex_complex(cot_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,X)),aa_complex_complex(sin_complex,X)) ).

% cot_def
tff(fact_8691_cot__def,axiom,
    ! [X: real] : aa_real_real(cot_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,X)) ).

% cot_def
tff(fact_8692_sinh__double,axiom,
    ! [Xb: complex] : aa_complex_complex(sinh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sinh_complex,Xb))),aa_complex_complex(cosh_complex,Xb)) ).

% sinh_double
tff(fact_8693_sinh__double,axiom,
    ! [Xb: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sinh_real,Xb))),aa_real_real(cosh_real,Xb)) ).

% sinh_double
tff(fact_8694_log2__of__power__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ).

% log2_of_power_eq
tff(fact_8695_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_8696_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),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,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8697_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_8698_tanh__add,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cosh_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,Xb)),aa_complex_complex(tanh_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,Xb)),aa_complex_complex(tanh_complex,Y)))) ) ) ) ).

% tanh_add
tff(fact_8699_tanh__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cosh_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Y) != zero_zero_real )
       => ( aa_real_real(tanh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y)))) ) ) ) ).

% tanh_add
tff(fact_8700_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_8701_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_8702_cosh__field__def,axiom,
    ! [Z: real] : aa_real_real(cosh_real,Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Z)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_8703_cosh__field__def,axiom,
    ! [Z: complex] : aa_complex_complex(cosh_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Z)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_8704_cosh__square__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% cosh_square_eq
tff(fact_8705_cosh__square__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% cosh_square_eq
tff(fact_8706_hyperbolic__pythagoras,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% hyperbolic_pythagoras
tff(fact_8707_hyperbolic__pythagoras,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_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(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% hyperbolic_pythagoras
tff(fact_8708_sinh__square__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% sinh_square_eq
tff(fact_8709_sinh__square__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% sinh_square_eq
tff(fact_8710_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_8711_cosh__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cosh_real,Xb) = zero_zero_real )
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_8712_cosh__zero__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cosh_complex,Xb) = zero_zero_complex )
    <=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_8713_cosh__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_8714_cosh__double,axiom,
    ! [Xb: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_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(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_8715_cosh__def,axiom,
    ! [Xb: real] : aa_real_real(cosh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% cosh_def
tff(fact_8716_cosh__def,axiom,
    ! [Xb: complex] : aa_complex_complex(cosh_complex,Xb) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).

% cosh_def
tff(fact_8717_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))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8718_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),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_8719_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_8720_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,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,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_8721_tan__cot_H,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).

% tan_cot'
tff(fact_8722_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_8723_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_8724_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_8725_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_8726_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_8727_of__int__ceiling__cancel,axiom,
    ! [Xb: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_ceiling_cancel
tff(fact_8728_of__int__ceiling__cancel,axiom,
    ! [Xb: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_ceiling_cancel
tff(fact_8729_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_8730_ceiling__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_8731_ceiling__one,axiom,
    archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_8732_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_8733_ceiling__add__of__int,axiom,
    ! [Xb: rat,Z: int] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),Z) ).

% ceiling_add_of_int
tff(fact_8734_ceiling__add__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z) ).

% ceiling_add_of_int
tff(fact_8735_ceiling__diff__of__int,axiom,
    ! [Xb: rat,Z: int] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),Z) ).

% ceiling_diff_of_int
tff(fact_8736_ceiling__diff__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z) ).

% ceiling_diff_of_int
tff(fact_8737_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_8738_ceiling__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_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_8739_zero__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% zero_less_ceiling
tff(fact_8740_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_8741_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),numeral_numeral_real(V)) ) ).

% ceiling_le_numeral
tff(fact_8742_ceiling__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),numeral_numeral_rat(V)) ) ).

% ceiling_le_numeral
tff(fact_8743_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_8744_ceiling__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_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_8745_one__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% one_le_ceiling
tff(fact_8746_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_8747_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,numeral_numeral_real(V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_8748_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)),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_8749_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_8750_ceiling__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_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_8751_one__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ).

% one_less_ceiling
tff(fact_8752_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_8753_ceiling__add__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_8754_ceiling__add__numeral,axiom,
    ! [Xb: rat,V: num] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_8755_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_8756_ceiling__neg__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_8757_ceiling__add__one,axiom,
    ! [Xb: rat] : archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_8758_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_8759_ceiling__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_8760_ceiling__diff__numeral,axiom,
    ! [Xb: rat,V: num] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_8761_ceiling__diff__one,axiom,
    ! [Xb: rat] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_8762_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_8763_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% ceiling_numeral_power
tff(fact_8764_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : archim2889992004027027881ng_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% ceiling_numeral_power
tff(fact_8765_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_8766_ceiling__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),uminus_uminus_rat(one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_8767_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_8768_zero__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),Xb) ) ).

% zero_le_ceiling
tff(fact_8769_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_8770_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,numeral_numeral_real(V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_8771_ceiling__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),minus_minus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_8772_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,numeral_numeral_real(V)),one_one_real)),Xb) ) ).

% numeral_le_ceiling
tff(fact_8773_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)),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),Xb) ) ).

% numeral_le_ceiling
tff(fact_8774_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,numeral_numeral_real(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_8775_ceiling__le__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_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),uminus_uminus_rat(numeral_numeral_rat(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_8776_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,numeral_numeral_real(V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_8777_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))),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_8778_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_8779_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,numeral_numeral_real(V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_8780_ceiling__less__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_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),minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_8781_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,numeral_numeral_real(V))),one_one_real)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_8782_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))),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_8783_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_8784_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,archim2889992004027027881ng_rat(Y)),archim2889992004027027881ng_rat(Xb)) ) ).

% ceiling_mono
tff(fact_8785_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_8786_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,archim2889992004027027881ng_rat(Xb))) ).

% le_of_int_ceiling
tff(fact_8787_ceiling__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),archim2889992004027027881ng_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_8788_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_8789_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_8790_ceiling__le__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_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_8791_ceiling__le,axiom,
    ! [Xb: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),A) ) ).

% ceiling_le
tff(fact_8792_ceiling__le,axiom,
    ! [Xb: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),A) ) ).

% ceiling_le
tff(fact_8793_less__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),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_8794_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_8795_ceiling__add__le,axiom,
    ! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),archim2889992004027027881ng_rat(Y))) ).

% ceiling_add_le
tff(fact_8796_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_8797_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_8798_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,archim2889992004027027881ng_rat(R2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,R2),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_8799_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_8800_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R2)),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_8801_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_8802_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : archim2889992004027027881ng_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A),aa_int_rat(ring_1_of_int_rat,B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_8803_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_8804_ceiling__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(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,archim2889992004027027881ng_rat(Xb))) ) ).

% ceiling_correct
tff(fact_8805_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_8806_ceiling__unique,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z))
       => ( archim2889992004027027881ng_rat(Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_8807_ceiling__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_8808_ceiling__eq__iff,axiom,
    ! [Xb: rat,A: int] :
      ( ( archim2889992004027027881ng_rat(Xb) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_8809_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_8810_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,archim2889992004027027881ng_rat(Ta))
    <=> ! [I3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I3),one_one_rat)),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_8811_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_8812_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B))) ) ) ).

% mult_ceiling_le
tff(fact_8813_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_8814_ceiling__less__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_8815_le__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xb) ) ).

% le_ceiling_iff
tff(fact_8816_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_8817_ceiling__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),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,P2),Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_8818_ceiling__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_8819_ceiling__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q3)))),one_one_real)),Q3)),P2) ) ).

% ceiling_divide_lower
tff(fact_8820_ceiling__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q3))),one_one_rat)),Q3)),P2) ) ).

% ceiling_divide_lower
tff(fact_8821_ceiling__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_8822_ceiling__eq,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Nb)),one_one_rat))
       => ( archim2889992004027027881ng_rat(Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_8823_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_8824_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K2)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K2))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8825_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8826_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8827_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_zd,Y)) ).

% arctan_def
tff(fact_8828_powr__one__eq__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,one_one_real),A) = one_one_real ).

% powr_one_eq_one
tff(fact_8829_of__int__floor__cancel,axiom,
    ! [Xb: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_floor_cancel
tff(fact_8830_of__int__floor__cancel,axiom,
    ! [Xb: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_floor_cancel
tff(fact_8831_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_8832_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_8833_floor__numeral,axiom,
    ! [V: num] : archim3151403230148437115or_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_8834_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_8835_floor__one,axiom,
    archim3151403230148437115or_rat(one_one_rat) = one_one_int ).

% floor_one
tff(fact_8836_powr__less__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_8837_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_8838_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_8839_powr__eq__one__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8840_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_8841_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_8842_powr__le__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_8843_numeral__powr__numeral__real,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,numeral_numeral_real(Mb)),numeral_numeral_real(Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_8844_floor__diff__of__int,axiom,
    ! [Xb: real,Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z) ).

% floor_diff_of_int
tff(fact_8845_floor__diff__of__int,axiom,
    ! [Xb: rat,Z: int] : archim3151403230148437115or_rat(minus_minus_rat(Xb,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(Xb)),Z) ).

% floor_diff_of_int
tff(fact_8846_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_8847_zero__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_floor
tff(fact_8848_zero__le__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb) ) ).

% zero_le_floor
tff(fact_8849_floor__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% floor_less_zero
tff(fact_8850_floor__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_8851_numeral__le__floor,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(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(V)),Xb) ) ).

% numeral_le_floor
tff(fact_8852_numeral__le__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(V)),Xb) ) ).

% numeral_le_floor
tff(fact_8853_zero__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% zero_less_floor
tff(fact_8854_zero__less__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).

% zero_less_floor
tff(fact_8855_floor__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_le_zero
tff(fact_8856_floor__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).

% floor_le_zero
tff(fact_8857_floor__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(V)) ) ).

% floor_less_numeral
tff(fact_8858_floor__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),numeral_numeral_rat(V)) ) ).

% floor_less_numeral
tff(fact_8859_one__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% one_le_floor
tff(fact_8860_one__le__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).

% one_le_floor
tff(fact_8861_floor__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_less_one
tff(fact_8862_floor__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).

% floor_less_one
tff(fact_8863_floor__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_8864_floor__neg__numeral,axiom,
    ! [V: num] : archim3151403230148437115or_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_8865_floor__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_8866_floor__diff__numeral,axiom,
    ! [Xb: rat,V: num] : archim3151403230148437115or_rat(minus_minus_rat(Xb,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_8867_floor__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_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(archim6058952711729229775r_real,Xb)),one_one_int) ).

% floor_diff_one
tff(fact_8868_floor__diff__one,axiom,
    ! [Xb: rat] : archim3151403230148437115or_rat(minus_minus_rat(Xb,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(Xb)),one_one_int) ).

% floor_diff_one
tff(fact_8869_floor__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% floor_numeral_power
tff(fact_8870_floor__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% floor_numeral_power
tff(fact_8871_powr__log__cancel,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_8872_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_8873_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_8874_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),numeral_numeral_real(Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_8875_numeral__less__floor,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(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)),Xb) ) ).

% numeral_less_floor
tff(fact_8876_numeral__less__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)),Xb) ) ).

% numeral_less_floor
tff(fact_8877_floor__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_8878_floor__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_8879_one__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),Xb) ) ).

% one_less_floor
tff(fact_8880_one__less__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(aa_num_num(bit0,one))),Xb) ) ).

% one_less_floor
tff(fact_8881_floor__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8882_floor__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),numeral_numeral_rat(aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_8883_neg__numeral__le__floor,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(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),Xb) ) ).

% neg_numeral_le_floor
tff(fact_8884_neg__numeral__le__floor,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))),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(V))),Xb) ) ).

% neg_numeral_le_floor
tff(fact_8885_floor__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_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_real,Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% floor_less_neg_numeral
tff(fact_8886_floor__less__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_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_rat,Xb),uminus_uminus_rat(numeral_numeral_rat(V))) ) ).

% floor_less_neg_numeral
tff(fact_8887_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_8888_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B))) = aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_8889_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_8890_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_8891_neg__numeral__less__floor,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(archim6058952711729229775r_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(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),Xb) ) ).

% neg_numeral_less_floor
tff(fact_8892_neg__numeral__less__floor,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))),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(V))),one_one_rat)),Xb) ) ).

% neg_numeral_less_floor
tff(fact_8893_floor__le__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_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_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_8894_floor__le__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_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_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_8895_square__powr__half,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% square_powr_half
tff(fact_8896_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_8897_powr__powr,axiom,
    ! [Xb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_8898_floor__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).

% floor_mono
tff(fact_8899_floor__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,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Y)) ) ).

% floor_mono
tff(fact_8900_of__int__floor__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(archim6058952711729229775r_real,Xb))),Xb) ).

% of_int_floor_le
tff(fact_8901_of__int__floor__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb))),Xb) ).

% of_int_floor_le
tff(fact_8902_floor__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% floor_less_cancel
tff(fact_8903_floor__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% floor_less_cancel
tff(fact_8904_powr__less__mono,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ) ) ).

% powr_less_mono
tff(fact_8905_powr__less__cancel,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_8906_powr__mono,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ) ) ).

% powr_mono
tff(fact_8907_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_8908_cis__divide,axiom,
    ! [A: real,B: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% cis_divide
tff(fact_8909_le__floor__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).

% le_floor_iff
tff(fact_8910_le__floor__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ) ).

% le_floor_iff
tff(fact_8911_floor__less__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).

% floor_less_iff
tff(fact_8912_floor__less__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim3151403230148437115or_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% floor_less_iff
tff(fact_8913_le__floor__add,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))) ).

% le_floor_add
tff(fact_8914_le__floor__add,axiom,
    ! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Y))),archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y))) ).

% le_floor_add
tff(fact_8915_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_8916_powr__inj,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_8917_floor__add__int,axiom,
    ! [Xb: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_int_real(ring_1_of_int_real,Z))) ).

% floor_add_int
tff(fact_8918_floor__add__int,axiom,
    ! [Xb: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),Z) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z))) ).

% floor_add_int
tff(fact_8919_int__add__floor,axiom,
    ! [Z: int,Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,Xb)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),Xb)) ).

% int_add_floor
tff(fact_8920_int__add__floor,axiom,
    ! [Z: int,Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),archim3151403230148437115or_rat(Xb)) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb)) ).

% int_add_floor
tff(fact_8921_powr__le1,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_8922_powr__mono__both,axiom,
    ! [A: real,B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),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),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_8923_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8924_powr__mult,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_8925_floor__divide__of__int__eq,axiom,
    ! [K2: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,K2)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(divide_divide_int(K2),L) ).

% floor_divide_of_int_eq
tff(fact_8926_floor__divide__of__int__eq,axiom,
    ! [K2: int,L: int] : archim3151403230148437115or_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K2),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(divide_divide_int(K2),L) ).

% floor_divide_of_int_eq
tff(fact_8927_floor__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,Xb)),Nb) ) ) ).

% floor_power
tff(fact_8928_floor__power,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb = aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb)) )
     => ( archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(Xb),Nb)) = aa_nat_int(power_power_int(archim3151403230148437115or_rat(Xb)),Nb) ) ) ).

% floor_power
tff(fact_8929_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_8930_ln__powr,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_8931_log__powr,axiom,
    ! [Xb: real,B: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_powr
tff(fact_8932_powr__add,axiom,
    ! [Xb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ).

% powr_add
tff(fact_8933_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z1),Z22)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z1)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z22)) ).

% powr_diff
tff(fact_8934_one__add__floor,axiom,
    ! [Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% one_add_floor
tff(fact_8935_one__add__floor,axiom,
    ! [Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),one_one_int) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)) ).

% one_add_floor
tff(fact_8936_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = K2 )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K2))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8937_floor__divide__of__nat__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_8938_floor__divide__of__nat__eq,axiom,
    ! [Mb: nat,Nb: nat] : archim3151403230148437115or_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb),aa_nat_rat(semiri681578069525770553at_rat,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_8939_powr__less__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_8940_less__powr__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_8941_log__less__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_less_iff
tff(fact_8942_less__log__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_8943_ceiling__altdef,axiom,
    ! [Xb: real] :
      aa_real_int(archim7802044766580827645g_real,Xb) = $ite(Xb = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)) ).

% ceiling_altdef
tff(fact_8944_ceiling__altdef,axiom,
    ! [Xb: rat] :
      archim2889992004027027881ng_rat(Xb) = $ite(Xb = aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb)),archim3151403230148437115or_rat(Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(Xb)),one_one_int)) ).

% ceiling_altdef
tff(fact_8945_ceiling__diff__floor__le__1,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim6058952711729229775r_real,Xb))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_8946_ceiling__diff__floor__le__1,axiom,
    ! [Xb: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),archim3151403230148437115or_rat(Xb))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_8947_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_8948_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_8949_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_8950_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_8951_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_8952_DeMoivre,axiom,
    ! [A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre
tff(fact_8953_floor__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,Ta))
    <=> ! [I3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I3)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I3)),one_one_real)) )
         => aa_int_o(P,I3) ) ) ).

% floor_split
tff(fact_8954_floor__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,archim3151403230148437115or_rat(Ta))
    <=> ! [I3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I3)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat)) )
         => aa_int_o(P,I3) ) ) ).

% floor_split
tff(fact_8955_floor__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),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,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_8956_floor__eq__iff,axiom,
    ! [Xb: rat,A: int] :
      ( ( archim3151403230148437115or_rat(Xb) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_8957_floor__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),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,Z)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Z ) ) ) ).

% floor_unique
tff(fact_8958_floor__unique,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
       => ( archim3151403230148437115or_rat(Xb) = Z ) ) ) ).

% floor_unique
tff(fact_8959_powr__minus__divide,axiom,
    ! [Xb: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ).

% powr_minus_divide
tff(fact_8960_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% le_mult_floor
tff(fact_8961_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B))),archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))) ) ) ).

% le_mult_floor
tff(fact_8962_less__floor__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% less_floor_iff
tff(fact_8963_less__floor__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim3151403230148437115or_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb) ) ).

% less_floor_iff
tff(fact_8964_floor__le__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),Z)
    <=> 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,Z)),one_one_real)) ) ).

% floor_le_iff
tff(fact_8965_floor__le__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim3151403230148437115or_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_8966_floor__correct,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(archim6058952711729229775r_real,Xb))),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,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int))) ) ).

% floor_correct
tff(fact_8967_floor__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(Xb))),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,archim3151403230148437115or_rat(Xb)),one_one_int))) ) ).

% floor_correct
tff(fact_8968_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_8969_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_8970_le__log__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_8971_log__le__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_le_iff
tff(fact_8972_le__powr__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_8973_powr__le__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_8974_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_8975_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(divide_divide_int(aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_8976_floor__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q3)))),Q3)),P2) ) ).

% floor_divide_lower
tff(fact_8977_floor__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P2,Q3)))),Q3)),P2) ) ).

% floor_divide_lower
tff(fact_8978_ln__powr__bound,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_8979_ln__powr__bound2,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xb)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_8980_log__add__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8981_add__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8982_minus__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8983_powr__def,axiom,
    ! [Xb: real,A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A) = $ite(Xb = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,Xb)))) ).

% powr_def
tff(fact_8984_floor__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q3)))),one_one_real)),Q3)) ) ).

% floor_divide_upper
tff(fact_8985_floor__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P2,Q3)))),one_one_rat)),Q3)) ) ).

% floor_divide_upper
tff(fact_8986_round__def,axiom,
    ! [Xb: real] : aa_real_int(archim8280529875227126926d_real,Xb) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8987_round__def,axiom,
    ! [Xb: rat] : archim7778729529865785530nd_rat(Xb) = archim3151403230148437115or_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% round_def
tff(fact_8988_log__minus__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8989_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),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_8990_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(Nb))) = 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,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_8991_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) = the_real(aTP_Lamm_ze) ).

% pi_half
tff(fact_8992_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),the_real(aTP_Lamm_ze)) ).

% pi_def
tff(fact_8993_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8994_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_zf,Y)) ).

% arcsin_def
tff(fact_8995_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => bij_betw_nat_complex(aTP_Lamm_zg(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cq,Nb))) ) ).

% bij_betw_roots_unity
tff(fact_8996_arcosh__def,axiom,
    ! [Xb: real] : 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(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_8997_round__altdef,axiom,
    ! [Xb: real] :
      aa_real_int(archim8280529875227126926d_real,Xb) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb),aa_real_int(archim6058952711729229775r_real,Xb)) ).

% round_altdef
tff(fact_8998_round__altdef,axiom,
    ! [Xb: rat] :
      archim7778729529865785530nd_rat(Xb) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),archimedean_frac_rat(Xb)),archim2889992004027027881ng_rat(Xb),archim3151403230148437115or_rat(Xb)) ).

% round_altdef
tff(fact_8999_The__split__eq,axiom,
    ! [Xb: code_integer,Y: $o] : the_Pr406357557219058217eger_o(produc7828578312038201481er_o_o(aa_o_f2459543391104474489un_o_o(aTP_Lamm_zh(Xb),(Y)))) = produc6677183202524767010eger_o(Xb,(Y)) ).

% The_split_eq
tff(fact_9000_The__split__eq,axiom,
    ! [Xb: num,Y: num] : the_Pr6395592806576110876um_num(produc5703948589228662326_num_o(aa_num5226579579167915582_num_o(aTP_Lamm_zi(Xb),Y))) = product_Pair_num_num(Xb,Y) ).

% The_split_eq
tff(fact_9001_The__split__eq,axiom,
    ! [Xb: nat,Y: num] : the_Pr8265262403268641490at_num(produc4927758841916487424_num_o(aa_num3446304307616776968_num_o(aTP_Lamm_zj(Xb),Y))) = product_Pair_nat_num(Xb,Y) ).

% The_split_eq
tff(fact_9002_The__split__eq,axiom,
    ! [Xb: nat,Y: nat] : the_Pr7557018466319803784at_nat(produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_zk(Xb),Y))) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y) ).

% The_split_eq
tff(fact_9003_The__split__eq,axiom,
    ! [Xb: int,Y: int] : the_Pr4378521158711661632nt_int(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_zl(Xb),Y))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y) ).

% The_split_eq
tff(fact_9004_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_9005_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_9006_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_9007_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_9008_of__real__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_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_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_mult
tff(fact_9009_of__real__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_mult
tff(fact_9010_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% of_real_numeral
tff(fact_9011_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,numeral_numeral_real(W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_9012_of__real__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_divide
tff(fact_9013_of__real__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_divide
tff(fact_9014_of__real__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,Xb)),Nb) ).

% of_real_power
tff(fact_9015_of__real__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),Nb) ).

% of_real_power
tff(fact_9016_of__real__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_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_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_9017_of__real__add,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_9018_of__real__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_diff
tff(fact_9019_of__real__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_diff
tff(fact_9020_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) ).

% of_real_neg_numeral
tff(fact_9021_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_9022_cos__of__real__pi,axiom,
    aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_of_real_pi
tff(fact_9023_cos__of__real__pi,axiom,
    aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cos_of_real_pi
tff(fact_9024_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_9025_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_9026_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_9027_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_9028_norm__of__real__addn,axiom,
    ! [Xb: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),numeral_numeral_real(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_9029_norm__of__real__addn,axiom,
    ! [Xb: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),numeral_numeral_real(B))) ).

% norm_of_real_addn
tff(fact_9030_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9031_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9032_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_9033_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_9034_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_9035_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_9036_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A3: complex,R4: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R4)),aa_complex_complex(exp_complex,A3)) ).

% complex_exp_exists
tff(fact_9037_scaleR__conv__of__real,axiom,
    ! [R2: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,R2),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V1803761363581548252l_real,R2)),Xb) ).

% scaleR_conv_of_real
tff(fact_9038_scaleR__conv__of__real,axiom,
    ! [R2: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),Xb) ).

% scaleR_conv_of_real
tff(fact_9039_of__real__def,axiom,
    ! [R2: real] : aa_real_real(real_V1803761363581548252l_real,R2) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,R2),one_one_real) ).

% of_real_def
tff(fact_9040_of__real__def,axiom,
    ! [R2: real] : aa_real_complex(real_V4546457046886955230omplex,R2) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),one_one_complex) ).

% of_real_def
tff(fact_9041_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_9042_frac__ge__0,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),archimedean_frac_rat(Xb)) ).

% frac_ge_0
tff(fact_9043_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_9044_frac__lt__1,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,archimedean_frac_rat(Xb)),one_one_rat) ).

% frac_lt_1
tff(fact_9045_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_9046_frac__1__eq,axiom,
    ! [Xb: rat] : archimedean_frac_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)) = archimedean_frac_rat(Xb) ).

% frac_1_eq
tff(fact_9047_nonzero__of__real__divide,axiom,
    ! [Y: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_9048_nonzero__of__real__divide,axiom,
    ! [Y: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_9049_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(aa_rea2649881322057434177omplex(complex2,Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_9050_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(aa_rea2649881322057434177omplex(complex2,Xb),Y)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_9051_cis__conv__exp,axiom,
    ! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% cis_conv_exp
tff(fact_9052_norm__less__p1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,Xb))),one_one_real))) ).

% norm_less_p1
tff(fact_9053_norm__less__p1,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Xb))),one_one_complex))) ).

% norm_less_p1
tff(fact_9054_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T2: set_nat,H: fun_int_nat,S3: set_int,T: set_nat,G: fun_nat_nat] :
      ( finite_finite_int(S4)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(bij_betw_int_nat(H,minus_minus_set_int(S3,S4)),minus_minus_set_nat(T,T2))
         => ( ! [A3: int] :
                ( member_int(A3,S4)
               => ( aa_nat_nat(G,aa_int_nat(H,A3)) = zero_zero_nat ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),T2)
                 => ( aa_nat_nat(G,B3) = zero_zero_nat ) )
             => ( groups4541462559716669496nt_nat(aa_fun6790803225114646484nt_nat(aTP_Lamm_zm(H),G),S3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9055_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_real,T2: set_nat,H: fun_real_nat,S3: set_real,T: set_nat,G: fun_nat_nat] :
      ( finite_finite_real(S4)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(bij_betw_real_nat(H,minus_minus_set_real(S3,S4)),minus_minus_set_nat(T,T2))
         => ( ! [A3: real] :
                ( member_real(A3,S4)
               => ( aa_nat_nat(G,aa_real_nat(H,A3)) = zero_zero_nat ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),T2)
                 => ( aa_nat_nat(G,B3) = zero_zero_nat ) )
             => ( groups1935376822645274424al_nat(aa_fun6786953957667191892al_nat(aTP_Lamm_zn(H),G),S3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9056_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_complex,T2: set_nat,H: fun_complex_nat,S3: set_complex,T: set_nat,G: fun_nat_nat] :
      ( finite3207457112153483333omplex(S4)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(bij_betw_complex_nat(H,minus_811609699411566653omplex(S3,S4)),minus_minus_set_nat(T,T2))
         => ( ! [A3: complex] :
                ( member_complex(A3,S4)
               => ( aa_nat_nat(G,aa_complex_nat(H,A3)) = zero_zero_nat ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),T2)
                 => ( aa_nat_nat(G,B3) = zero_zero_nat ) )
             => ( groups5693394587270226106ex_nat(aa_fun8118263755821012566ex_nat(aTP_Lamm_zo(H),G),S3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9057_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T2: set_nat,H: fun_int_nat,S3: set_int,T: set_nat,G: fun_nat_real] :
      ( finite_finite_int(S4)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(bij_betw_int_nat(H,minus_minus_set_int(S3,S4)),minus_minus_set_nat(T,T2))
         => ( ! [A3: int] :
                ( member_int(A3,S4)
               => ( aa_nat_real(G,aa_int_nat(H,A3)) = zero_zero_real ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),T2)
                 => ( aa_nat_real(G,B3) = zero_zero_real ) )
             => ( groups8778361861064173332t_real(aa_fun4026127921582108556t_real(aTP_Lamm_zp(H),G),S3) = groups6591440286371151544t_real(G,T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9058_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_real,T2: set_nat,H: fun_real_nat,S3: set_real,T: set_nat,G: fun_nat_real] :
      ( finite_finite_real(S4)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(bij_betw_real_nat(H,minus_minus_set_real(S3,S4)),minus_minus_set_nat(T,T2))
         => ( ! [A3: real] :
                ( member_real(A3,S4)
               => ( aa_nat_real(G,aa_real_nat(H,A3)) = zero_zero_real ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),T2)
                 => ( aa_nat_real(G,B3) = zero_zero_real ) )
             => ( groups8097168146408367636l_real(aa_fun6767034839249511948l_real(aTP_Lamm_zq(H),G),S3) = groups6591440286371151544t_real(G,T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9059_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_complex,T2: set_nat,H: fun_complex_nat,S3: set_complex,T: set_nat,G: fun_nat_real] :
      ( finite3207457112153483333omplex(S4)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(bij_betw_complex_nat(H,minus_811609699411566653omplex(S3,S4)),minus_minus_set_nat(T,T2))
         => ( ! [A3: complex] :
                ( member_complex(A3,S4)
               => ( aa_nat_real(G,aa_complex_nat(H,A3)) = zero_zero_real ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),T2)
                 => ( aa_nat_real(G,B3) = zero_zero_real ) )
             => ( groups5808333547571424918x_real(aa_fun2896631202927662350x_real(aTP_Lamm_zr(H),G),S3) = groups6591440286371151544t_real(G,T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9060_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T2: set_real,H: fun_int_real,S3: set_int,T: set_real,G: fun_real_int] :
      ( finite_finite_int(S4)
     => ( finite_finite_real(T2)
       => ( bij_betw_int_real(H,minus_minus_set_int(S3,S4),minus_minus_set_real(T,T2))
         => ( ! [A3: int] :
                ( member_int(A3,S4)
               => ( aa_real_int(G,aa_int_real(H,A3)) = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( member_real(B3,T2)
                 => ( aa_real_int(G,B3) = zero_zero_int ) )
             => ( groups4538972089207619220nt_int(aa_fun4720285486439830120nt_int(aTP_Lamm_zs(H),G),S3) = groups1932886352136224148al_int(G,T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9061_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T2: set_complex,H: fun_int_complex,S3: set_int,T: set_complex,G: fun_complex_int] :
      ( finite_finite_int(S4)
     => ( finite3207457112153483333omplex(T2)
       => ( bij_betw_int_complex(H,minus_minus_set_int(S3,S4),minus_811609699411566653omplex(T,T2))
         => ( ! [A3: int] :
                ( member_int(A3,S4)
               => ( aa_complex_int(G,aa_int_complex(H,A3)) = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( member_complex(B3,T2)
                 => ( aa_complex_int(G,B3) = zero_zero_int ) )
             => ( groups4538972089207619220nt_int(aa_fun4384185347138573674nt_int(aTP_Lamm_zt(H),G),S3) = groups5690904116761175830ex_int(G,T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9062_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T2: set_nat,H: fun_int_nat,S3: set_int,T: set_nat,G: fun_nat_int] :
      ( finite_finite_int(S4)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(bij_betw_int_nat(H,minus_minus_set_int(S3,S4)),minus_minus_set_nat(T,T2))
         => ( ! [A3: int] :
                ( member_int(A3,S4)
               => ( aa_nat_int(G,aa_int_nat(H,A3)) = zero_zero_int ) )
           => ( ! [B3: nat] :
                  ( aa_set_nat_o(member_nat(B3),T2)
                 => ( aa_nat_int(G,B3) = zero_zero_int ) )
             => ( groups4538972089207619220nt_int(aa_fun6564396834021159052nt_int(aTP_Lamm_zu(H),G),S3) = groups3539618377306564664at_int(G,T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9063_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T2: set_int,H: fun_int_int,S3: set_int,T: set_int,G: fun_int_int] :
      ( finite_finite_int(S4)
     => ( finite_finite_int(T2)
       => ( bij_betw_int_int(H,minus_minus_set_int(S3,S4),minus_minus_set_int(T,T2))
         => ( ! [A3: int] :
                ( member_int(A3,S4)
               => ( aa_int_int(G,aa_int_int(H,A3)) = zero_zero_int ) )
           => ( ! [B3: int] :
                  ( member_int(B3,T2)
                 => ( aa_int_int(G,B3) = zero_zero_int ) )
             => ( groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_zv(H),G),S3) = groups4538972089207619220nt_int(G,T) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_9064_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_9065_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_9066_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(aa_rea2649881322057434177omplex(complex2,A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_9067_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R4: real,A3: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R4)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A3))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A3))))) ).

% complex_split_polar
tff(fact_9068_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aTP_Lamm_zw(Xb)) ).

% floor_real_def
tff(fact_9069_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_9070_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9071_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9072_modulo__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Mb)),
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).

% modulo_int_unfold
tff(fact_9073_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_9074_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_9075_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_9076_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_9077_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9078_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9079_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K2)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9080_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K2)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9081_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9082_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_9083_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N: nat,L3: int] : K2 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_9084_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_9085_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9086_sgn__mod,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K2),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9087_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_9088_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_9089_div__sgn__abs__cancel,axiom,
    ! [V: int,K2: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_9090_div__dvd__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
     => ( aa_int_int(divide_divide_int(K2),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_int_int(sgn_sgn_int,L))),aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_9091_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9092_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_9093_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_9094_div__noneq__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K2) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(divide_divide_int(K2),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K2)),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),K2))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9095_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q3: 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))
       => ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9096_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_9097_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(divide_divide_int(aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_9098_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_9099_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9100_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q2: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22) ) ) )
         => ~ ! [R4: int,Q2: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R4) )
               => ( ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R4)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9101_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_9102_divide__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_9103_floor__rat__def,axiom,
    ! [Xb: rat] : archim3151403230148437115or_rat(Xb) = the_int(aTP_Lamm_zx(Xb)) ).

% floor_rat_def
tff(fact_9104_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cq,Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_zy(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9105_divide__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(divide_divide_int(K2),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = 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,K2))),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,K2))),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),K2)))))) ) ).

% divide_int_def
tff(fact_9106_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_9107_nat__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_numeral
tff(fact_9108_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_9109_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9110_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9111_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9112_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_9113_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_9114_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_9115_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9116_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9117_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9118_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9119_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9120_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9121_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_9122_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9123_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9124_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9125_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9126_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_9127_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9128_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9129_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_9130_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_9131_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9132_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_9133_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9134_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9135_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9136_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9137_nat__mask__eq,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% nat_mask_eq
tff(fact_9138_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9139_sgn__rat__def,axiom,
    ! [A: rat] :
      sgn_sgn_rat(A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9140_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)
           => ! [T3: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T3)
               => ( R2 != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S),T3) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9141_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_mult
tff(fact_9142_real__root__mult__exp,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Xb) = aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_mult_exp
tff(fact_9143_nat__numeral__as__int,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,X) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X)) ).

% nat_numeral_as_int
tff(fact_9144_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_9145_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9146_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_9147_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(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)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_9148_unset__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(bit_se4203085406695923979it_int(Mb),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_9149_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [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)),Nb)) = Xb )
             => aa_real_o(P,Y5) ) ) ) ) ).

% split_root
tff(fact_9150_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_9151_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_9152_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),K2) ) ) ).

% real_root_power
tff(fact_9153_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_9154_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_9155_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9156_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_9157_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_9158_int__minus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)))) ).

% int_minus
tff(fact_9159_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9160_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_9161_nat__times__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_9162_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_9163_nat__minus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_9164_nat__div__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_9165_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9166_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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,Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9167_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9168_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9169_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9170_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_9171_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_9172_le__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K2) ) ) ).

% le_nat_iff
tff(fact_9173_nat__add__distrib,axiom,
    ! [Z: int,Z5: 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),Z5)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_9174_nat__mult__distrib,axiom,
    ! [Z: int,Z5: 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),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_9175_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_9176_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z5),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_9177_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_9178_nat__abs__triangle__ineq,axiom,
    ! [K2: 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,K2),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_9179_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(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_9180_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(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_9181_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_9182_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_9183_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(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_9184_div__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_9185_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_9186_le__nat__floor,axiom,
    ! [Xb: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_9187_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9188_real__root__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N3),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9189_real__root__decreasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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,Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9190_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9191_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9192_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9193_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_9194_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9195_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_9196_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_9197_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xb: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xb )
     => ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9198_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_9199_nat__less__iff,axiom,
    ! [W: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),Mb)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).

% nat_less_iff
tff(fact_9200_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: 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),Z5)) = 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,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9201_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_9202_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_9203_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5),
          $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_9204_real__root__increasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N3),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9205_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9206_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9207_log__base__root,axiom,
    ! [Nb: nat,B: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% log_base_root
tff(fact_9208_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9209_even__nat__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( 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,K2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ) ).

% even_nat_iff
tff(fact_9210_modulo__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(modulo_modulo_int(K2),L) = $ite(
        L = zero_zero_int,
        K2,
        $ite(aa_int_int(sgn_sgn_int,K2) = 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,K2))),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),K2)))),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,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_9211_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xb)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_9212_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K2),L) = $ite(
        ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K2 = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_9213_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_concat_bit(Nb,K2),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(Nb),K2) ).

% concat_bit_of_zero_2
tff(fact_9214_and__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_9215_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_9216_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_9217_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_9218_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_9219_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9220_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9221_diff__rat__def,axiom,
    ! [Q3: rat,R2: rat] : minus_minus_rat(Q3,R2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Q3),uminus_uminus_rat(R2)) ).

% diff_rat_def
tff(fact_9222_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9223_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_9224_nat__take__bit__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K2)) ) ) ).

% nat_take_bit_eq
tff(fact_9225_take__bit__nat__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ) ).

% take_bit_nat_eq
tff(fact_9226_take__bit__diff,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)) ).

% take_bit_diff
tff(fact_9227_take__bit__mult,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K2),L)) ).

% take_bit_mult
tff(fact_9228_take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K2)) ).

% take_bit_minus
tff(fact_9229_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B: int] : bit_concat_bit(Nb,aa_int_int(bit_se2923211474154528505it_int(Nb),B)) = bit_concat_bit(Nb,B) ).

% concat_bit_take_bit_eq
tff(fact_9230_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K2: int,L: int,R2: int,S2: int] :
      ( ( aa_int_int(bit_concat_bit(Nb,K2),L) = aa_int_int(bit_concat_bit(Nb,R2),S2) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_int_int(bit_se2923211474154528505it_int(Nb),R2) )
        & ( L = S2 ) ) ) ).

% concat_bit_eq_iff
tff(fact_9231_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Mb),K2)),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9232_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(bit_se725231765392027082nd_int(Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_9233_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_9234_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(bit_se725231765392027082nd_int(Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_9235_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(bit_se725231765392027082nd_int(Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_9236_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(bit_se725231765392027082nd_int(Xb),Y)) ) ).

% AND_lower
tff(fact_9237_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9238_take__bit__nonnegative,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% take_bit_nonnegative
tff(fact_9239_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(bit_se2923211474154528505it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_9240_not__take__bit__negative,axiom,
    ! [Nb: nat,K2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_9241_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),one) = Xb ).

% pow.simps(1)
tff(fact_9242_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(bit_se725231765392027082nd_int(Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_9243_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_9244_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K2),L)),K2) ) ).

% and_less_eq
tff(fact_9245_take__bit__decr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9246_even__and__iff__int,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_9247_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9248_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9249_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_9250_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_9251_take__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_9252_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_9253_take__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_int_int(modulo_modulo_int(K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_9254_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_9255_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_9256_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(bit_se2923211474154528505it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9257_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% take_bit_int_less_self_iff
tff(fact_9258_take__bit__int__eq__self,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_9259_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9260_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_9261_take__bit__incr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ) ).

% take_bit_incr_eq
tff(fact_9262_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_9263_take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),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,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ).

% take_bit_int_greater_eq
tff(fact_9264_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9265_and__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K2),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))),K2)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_9266_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9267_take__bit__minus__small__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9268_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_9269_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K2),L) = $ite(
        ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & member_int(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))),K2)
            & ~ 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))),K2)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_9270_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
     => ( Y = $ite(
            ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xaa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9271_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_9272_pred__numeral__inc,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,inc(K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% pred_numeral_inc
tff(fact_9273_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_9274_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_9275_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_9276_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_9277_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9278_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9279_num__induct,axiom,
    ! [P: fun_num_o,Xb: num] :
      ( aa_num_o(P,one)
     => ( ! [X4: num] :
            ( aa_num_o(P,X4)
           => aa_num_o(P,inc(X4)) )
       => aa_num_o(P,Xb) ) ) ).

% num_induct
tff(fact_9280_add__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),inc(Y)) = inc(aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),Y)) ).

% add_inc
tff(fact_9281_inc_Osimps_I1_J,axiom,
    inc(one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9282_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : inc(aa_num_num(bit1,Xb)) = aa_num_num(bit0,inc(Xb)) ).

% inc.simps(3)
tff(fact_9283_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : inc(aa_num_num(bit0,Xb)) = aa_num_num(bit1,Xb) ).

% inc.simps(2)
tff(fact_9284_add__One,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),one) = inc(Xb) ).

% add_One
tff(fact_9285_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_9286_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_9287_and__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_9288_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xb),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_9289_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb))
     => ( set_or1266510415728281911st_int(Mb,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),set_or1266510415728281911st_int(Mb,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_9290_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_9291_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_9292_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9293_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L))
     => ( aa_int_int(bit_se725231765392027082nd_int(K2),L) = $ite(
            ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(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))),K2)
                & ~ 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))),K2)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_9294_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xaa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9295_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9296_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9297_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_9298_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9299_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9300_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9301_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9302_set__encode__insert,axiom,
    ! [A4: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Nb),A4)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_9303_bit__and__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se725231765392027082nd_int(K2),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_9304_lessThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2)) = insert_nat(K2,aa_nat_set_nat(set_ord_lessThan_nat,K2)) ).

% lessThan_Suc
tff(fact_9305_atMost__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2)) = insert_nat(aa_nat_nat(suc,K2),aa_nat_set_nat(set_ord_atMost_nat,K2)) ).

% atMost_Suc
tff(fact_9306_bit__not__int__iff_H,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K2)),one_one_int)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% bit_not_int_iff'
tff(fact_9307_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_9308_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) = insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_9309_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_9310_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_9311_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K2))) ).

% lessThan_nat_numeral
tff(fact_9312_atMost__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K2)) = insert_nat(aa_num_nat(numeral_numeral_nat,K2),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K2))) ).

% atMost_nat_numeral
tff(fact_9313_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Mb),K2)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9314_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(Mb,K2),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9315_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(bit_concat_bit(Nb,K2),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9316_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K2),M2)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K2),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),N) ) ) ) ).

% int_bit_bound
tff(fact_9317_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9318_bit__int__def,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_int_def
tff(fact_9319_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Z)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9320_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L3))
           => ( ( ~ ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9321_set__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% set_bit_eq
tff(fact_9322_unset__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_9323_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% take_bit_Suc_from_most
tff(fact_9324_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_9325_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K2),L) = $ite(
        ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K2 = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ord_max_int(aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9326_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9327_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_zz),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9328_or__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_9329_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9330_quotient__of__number_I3_J,axiom,
    ! [K2: num] : quotient_of(numeral_numeral_rat(K2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9331_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_9332_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_9333_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_9334_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_9335_quotient__of__number_I5_J,axiom,
    ! [K2: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K2))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9336_quotient__of__number_I4_J,axiom,
    quotient_of(uminus_uminus_rat(one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9337_bit__or__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se1409905431419307370or_int(K2),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
        | aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_9338_divide__rat__def,axiom,
    ! [Q3: rat,R2: rat] : divide_divide_rat(Q3,R2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Q3),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_9339_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9340_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_9341_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(bit_se1409905431419307370or_int(Xb),Y)) ) ) ).

% OR_lower
tff(fact_9342_or__greater__eq,axiom,
    ! [L: int,K2: 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,K2),aa_int_int(bit_se1409905431419307370or_int(K2),L)) ) ).

% or_greater_eq
tff(fact_9343_plus__and__or,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),aa_int_int(bit_se1409905431419307370or_int(Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) ).

% plus_and_or
tff(fact_9344_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_9345_bit__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K2)),Nb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ) ).

% bit_nat_iff
tff(fact_9346_rat__less__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_aab(Q3)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9347_rat__less__eq__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_aad(Q3)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9348_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),pi)) = zero_zero_real )
    <=> member_real(Xb,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9349_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_9350_OR__upper,axiom,
    ! [Xb: int,Nb: 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))),Nb))
       => ( 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))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_9351_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9352_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9353_or__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K2),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))),K2)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9354_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),one_one_int) ).

% quotient_of_int
tff(fact_9355_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(minus_minus_rat(P2,Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aaf(Q3)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9356_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aah(Q3)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9357_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(Nb)))) ).

% or_minus_numerals(5)
tff(fact_9358_normalize__denom__zero,axiom,
    ! [P2: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9359_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_9360_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_9361_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_9362_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_9363_or__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_9364_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_9365_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),bitM(Nb)))) ).

% or_minus_numerals(7)
tff(fact_9366_or__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),bitM(Nb)))) ).

% or_minus_numerals(3)
tff(fact_9367_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(Nb)))) ).

% or_minus_numerals(1)
tff(fact_9368_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9369_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9370_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(6)
tff(fact_9371_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(3)
tff(fact_9372_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9373_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit0,Mb)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(5)
tff(fact_9374_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit1,Mb)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(9)
tff(fact_9375_or__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9376_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(2)
tff(fact_9377_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit0,Mb)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(8)
tff(fact_9378_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P2: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q3)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9379_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xb = one )
           => ! [M: num] :
                ( ( Xaa = aa_num_num(bit0,M) )
               => ( Y != aa_num_num(bit1,M) ) ) )
         => ( ( ( Xb = one )
             => ! [M: num] :
                  ( ( Xaa = aa_num_num(bit1,M) )
                 => ( Y != aa_num_num(bit1,M) ) ) )
           => ( ( ? [N: num] : Xb = aa_num_num(bit0,N)
               => ( ( Xaa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa_num_num(bit0,N) )
                   => ! [M: num] :
                        ( ( Xaa = aa_num_num(bit0,M) )
                       => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa_num_num(bit0,N) )
                     => ! [M: num] :
                          ( ( Xaa = aa_num_num(bit1,M) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
                 => ( ( ? [N: num] : Xb = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa_num_num(bit1,N) )
                         => ! [M: num] :
                              ( ( Xaa = aa_num_num(bit0,M) )
                             => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xb = aa_num_num(bit1,N) )
                           => ! [M: num] :
                                ( ( Xaa = aa_num_num(bit1,M) )
                               => ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9380_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_9381_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_9382_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9383_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(divide_divide_rat(P2,Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aaj(Q3)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9384_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),Q3)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aal(Q3)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9385_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9386_Frct__code__post_I5_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K2))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K2)) ).

% Frct_code_post(5)
tff(fact_9387_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9388_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9389_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_9390_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_9391_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_9392_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_9393_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_9394_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9395_Frct__code__post_I4_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int)) = numeral_numeral_rat(K2) ).

% Frct_code_post(4)
tff(fact_9396_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9397_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9398_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9399_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9400_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9401_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(bit_concat_bit(Nb,zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(Nb),L) ).

% concat_bit_of_zero_1
tff(fact_9402_xor__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9403_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K2),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_9404_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9405_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9406_flip__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2159334234014336723it_int(Nb),K2) = aa_int_int(bit_se6526347334894502574or_int(K2),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% flip_bit_int_def
tff(fact_9407_bit__xor__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se6526347334894502574or_int(K2),L)),Nb)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
        <=> aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_9408_push__bit__nat__eq,axiom,
    ! [Nb: nat,K2: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(Nb),K2)) ).

% push_bit_nat_eq
tff(fact_9409_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less_eq
tff(fact_9410_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less_eq
tff(fact_9411_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(bit_se6526347334894502574or_int(Xb),Y)) ) ) ).

% XOR_lower
tff(fact_9412_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9413_set__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9414_flip__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9415_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_9416_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Mb),K2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9417_xor__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9418_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q3)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9419_concat__bit__eq,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_concat_bit(Nb,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_eq
tff(fact_9420_concat__bit__def,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_concat_bit(Nb,K2),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_def
tff(fact_9421_set__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),K2) = aa_int_int(bit_se1409905431419307370or_int(K2),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% set_bit_int_def
tff(fact_9422_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9423_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9424_prod__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9425_push__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se545348938243370406it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_int_def
tff(fact_9426_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9427_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K2: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(numeral_numeral_nat,K2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K2)),insert_nat(aa_num_nat(pred_numeral,K2),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(pred_numeral,K2))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9428_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_minus_one
tff(fact_9429_XOR__upper,axiom,
    ! [Xb: int,Nb: 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))),Nb))
       => ( 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))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9430_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9431_xor__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K2),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))),K2) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9432_sum__power2,axiom,
    ! [K2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K2)) = 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))),K2)),one_one_nat) ).

% sum_power2
tff(fact_9433_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pp(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9434_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K2),L) = $ite(
        K2 = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K2),
          $ite(
            K2 = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9435_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% not_negative_int_iff
tff(fact_9436_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9437_and__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9438_or__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9439_bit__not__int__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K2)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% bit_not_int_iff
tff(fact_9440_or__int__def,axiom,
    ! [K2: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K2),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K2)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9441_not__int__def,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K2)),one_one_int) ).

% not_int_def
tff(fact_9442_and__not__numerals_I1_J,axiom,
    aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9443_or__not__numerals_I1_J,axiom,
    aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9444_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_9445_unset__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),K2) = aa_int_int(bit_se725231765392027082nd_int(K2),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int))) ).

% unset_bit_int_def
tff(fact_9446_xor__int__def,axiom,
    ! [K2: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K2),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(K2),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K2)),L)) ).

% xor_int_def
tff(fact_9447_not__int__div__2,axiom,
    ! [K2: int] : aa_int_int(divide_divide_int(aa_int_int(bit_ri7919022796975470100ot_int,K2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9448_even__not__iff__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K2))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_not_iff_int
tff(fact_9449_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9450_and__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(4)
tff(fact_9451_or__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9452_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9453_bit__minus__int__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K2)),Nb)
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9454_numeral__or__not__num__eq,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9455_int__numeral__not__or__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb))) ).

% int_numeral_not_or_num_neg
tff(fact_9456_int__numeral__or__not__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9457_and__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9458_and__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(7)
tff(fact_9459_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9460_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9461_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9462_and__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9463_and__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9464_or__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9465_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9466_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9467_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9468_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9469_not__int__rec,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = 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))),K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9470_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_9471_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M5)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X6,M5)),aa_nat_real(X6,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9472_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_9473_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_9474_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_9475_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu2),(Uv2))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9476_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_9477_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9478_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_gv(L),Qr) ).

% divmod_step_integer_def
tff(fact_9479_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9480_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9481_minus__integer__code_I1_J,axiom,
    ! [K2: code_integer] : minus_8373710615458151222nteger(K2,zero_z3403309356797280102nteger) = K2 ).

% minus_integer_code(1)
tff(fact_9482_sgn__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = $ite(
        K2 = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9483_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9484_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_9485_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_9486_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(re,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb)) ).

% scaleR_complex.simps(1)
tff(fact_9487_minus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y)) ).

% minus_complex.simps(1)
tff(fact_9488_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_9489_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9490_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_9491_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
            $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9492_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9493_integer__of__int__code,axiom,
    ! [K2: int] :
      aa_int_Code_integer(code_integer_of_int,K2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K2))),
        $ite(
          K2 = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9494_Im__i__times,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_complex_real(re,Z) ).

% Im_i_times
tff(fact_9495_Re__i__times,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% Re_i_times
tff(fact_9496_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xb)),zero_zero_real)
        | ( ( aa_complex_real(im,Xb) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xb)) ) ) ).

% csqrt_minus
tff(fact_9497_csqrt__of__real__nonpos,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xb) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9498_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_9499_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9500_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_9501_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_9502_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : 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_9503_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(im,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb)) ).

% scaleR_complex.simps(2)
tff(fact_9504_minus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)) ).

% minus_complex.simps(2)
tff(fact_9505_times__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_9506_times__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_9507_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb))) ).

% scaleR_complex.code
tff(fact_9508_minus__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(minus_minus_complex(Xb),Y) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% minus_complex.code
tff(fact_9509_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_9510_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_9511_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_9512_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_9513_times__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(Xb),Y) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_9514_exp__eq__polar,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,Z) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z)))),aa_real_complex(cis,aa_complex_real(im,Z))) ).

% exp_eq_polar
tff(fact_9515_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9516_Im__power2,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,Xb))),aa_complex_real(im,Xb)) ).

% Im_power2
tff(fact_9517_Re__power2,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9518_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9519_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9520_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9521_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_9522_Re__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9523_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_9524_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9525_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9526_Im__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9527_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_9528_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9529_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa_complex_complex(invers8013647133539491842omplex,Xb) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9530_Complex__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_real_complex(aa_rea2649881322057434177omplex(complex2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9531_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9532_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9533_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member_complex(Y,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( Xb = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9534_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member_complex(Y,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = Xb )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9535_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9536_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9537_complex__cnj__mult,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_mult
tff(fact_9538_complex__cnj__one,axiom,
    aa_complex_complex(cnj,one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_9539_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(cnj,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
tff(fact_9540_complex__cnj__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_diff
tff(fact_9541_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9542_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_9543_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_9544_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9545_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_gt_0
tff(fact_9546_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_9547_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_9548_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_ge_0
tff(fact_9549_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_gt_0
tff(fact_9550_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_9551_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_9552_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_ge_0
tff(fact_9553_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9554_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ) ).

% complex_div_gt_0
tff(fact_9555_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9556_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9557_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9558_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9559_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit1,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9560_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_int_of_integer(K2) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2))),
        $ite(K2 = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_aam,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9561_num__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_num_of_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_aan,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9562_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_9563_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9564_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(minus_8373710615458151222nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% minus_integer.rep_eq
tff(fact_9565_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9566_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9567_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(aa_num_num(bit0,one)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9568_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_nat_of_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_aao,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9569_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)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      some_nat(one_one_nat),
                      $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [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_9570_card__Collect__less__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_9571_card__atMost,axiom,
    ! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_9572_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_9573_card__Collect__le__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bt,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9574_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9575_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_9576_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_9577_card__less__Suc2,axiom,
    ! [M6: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aap(M6),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aaq(M6),I))) ) ) ).

% card_less_Suc2
tff(fact_9578_card__less__Suc,axiom,
    ! [M6: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aap(M6),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aaq(M6),I))) ) ) ).

% card_less_Suc
tff(fact_9579_card__less,axiom,
    ! [M6: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aaq(M6),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9580_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),aa_nat_set_nat(set_or4665077453230672383an_nat(K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),finite_card_nat(A4))))
     => ( A4 = aa_nat_set_nat(set_or4665077453230672383an_nat(K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),finite_card_nat(A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9581_nat__of__integer__code__post_I3_J,axiom,
    ! [K2: num] : code_nat_of_integer(numera6620942414471956472nteger(K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_of_integer_code_post(3)
tff(fact_9582_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9583_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(N3)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9584_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),S3)) ).

% card_sum_le_nat_sum
tff(fact_9585_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_zy(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9586_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cq,Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9587_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)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      (A3),
                      some_nat(zero_zero_nat),
                      $ite((B3),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [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_9588_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)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu: $o] :
                  ( ( Xb = vEBT_Leaf((Uu),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9589_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)
       => ( ! [Uv: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) )
         => ( ! [Uu: $o] :
                ( ( Xb = vEBT_Leaf((Uu),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9590_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)) )
         => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9591_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_aar),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9592_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_aas),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9593_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9594_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_aat,Nb)),Mb) ).

% max_Suc2
tff(fact_9595_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_aau,Nb)),Mb) ).

% max_Suc1
tff(fact_9596_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_an),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_9597_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_9598_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_an),Nat) ).

% pred_def
tff(fact_9599_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_9600_prod__decode__aux_Osimps,axiom,
    ! [K2: nat,Mb: nat] :
      nat_prod_decode_aux(K2,Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),Mb)),nat_prod_decode_aux(aa_nat_nat(suc,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K2)))) ).

% prod_decode_aux.simps
tff(fact_9601_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K2)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9602_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9603_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9604_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9605_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9606_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9607_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K2)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9608_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K2: int] : aa_int_int(bit_se8568078237143864401it_int(Mb),aa_int_int(bit_se545348938243370406it_int(Nb),K2)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),K2)) ).

% drop_bit_push_bit_int
tff(fact_9609_drop__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K2) = aa_int_int(divide_divide_int(K2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_int_def
tff(fact_9610_Suc__0__div__numeral,axiom,
    ! [K2: 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,K2)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_div_numeral
tff(fact_9611_Suc__0__mod__numeral,axiom,
    ! [K2: 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,K2)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_mod_numeral
tff(fact_9612_bit__cut__integer__code,axiom,
    ! [K2: code_integer] :
      code_bit_cut_integer(K2) = $ite(K2 = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_aav(K2),code_divmod_abs(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9613_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9614_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ).

% fst_divmod_nat
tff(fact_9615_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K2: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(Nb),K2)) ).

% drop_bit_nat_eq
tff(fact_9616_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9617_bit__cut__integer__def,axiom,
    ! [K2: code_integer] : code_bit_cut_integer(K2) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K2)) ).

% bit_cut_integer_def
tff(fact_9618_divmod__integer__code,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = 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),K2),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_aaw(L),code_divmod_abs(K2,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_aax(L),code_divmod_abs(K2,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9619_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) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_non_0
tff(fact_9620_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,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_9621_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,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_9622_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(sgn_sgn_rat(P2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,product_fst_int_int(quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_9623_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb)))) ).

% one_mod_minus_numeral
tff(fact_9624_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_mod_numeral
tff(fact_9625_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,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,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_9626_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      aa_int_int(adjust_mod(L),R2) = $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2)) ).

% Divides.adjust_mod_def
tff(fact_9627_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_9628_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P2)),
        $let(
          a: int,
          a:= aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(product_fst_int_int(P2)),a)),aa_int_int(divide_divide_int(product_snd_int_int(P2)),a)) ),
        $ite(
          product_snd_int_int(P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(gcd_gcd_int(product_fst_int_int(P2)),product_snd_int_int(P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(divide_divide_int(product_fst_int_int(P2)),a)),aa_int_int(divide_divide_int(product_snd_int_int(P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9629_gcd__1__int,axiom,
    ! [Mb: int] : aa_int_int(gcd_gcd_int(Mb),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9630_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y)) = aa_int_int(gcd_gcd_int(Xb),Y) ).

% bezout_int
tff(fact_9631_gcd__mult__distrib__int,axiom,
    ! [K2: int,Mb: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K2)),aa_int_int(gcd_gcd_int(Mb),Nb)) = aa_int_int(gcd_gcd_int(aa_int_int(aa_int_fun_int_int(times_times_int,K2),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9632_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Mb: nat] :
      ( ! [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
         => aa_nat_o(P,K) )
     => ( ! [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),I4)
                 => aa_nat_o(P,I4) )
             => aa_nat_o(P,K) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_9633_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_9634_gcd__1__nat,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9635_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9636_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb))
    <=> ( ( Mb != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9637_gcd__mult__distrib__nat,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9638_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9639_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9640_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9641_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9642_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9643_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: 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,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9644_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_9645_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9646_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_9647_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_9648_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9649_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9650_tanh__real__bounds,axiom,
    ! [Xb: real] : member_real(aa_real_real(tanh_real,Xb),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_9651_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9652_unbounded__k__infinite,axiom,
    ! [K2: nat,S3: set_nat] :
      ( ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),M)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N7)
              & aa_set_nat_o(member_nat(N7),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_9653_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [R4: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R4),aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S3)))
          & ! [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N7),finite_card_nat(S3))
             => aa_set_nat_o(member_nat(aa_nat_nat(R4,N7)),S3) ) ) ) ).

% finite_enumerate
tff(fact_9654_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K2: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K2) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(neg_numeral_sub_int(Nb,one)),K2)) ).

% xor_minus_numerals(1)
tff(fact_9655_xor__minus__numerals_I2_J,axiom,
    ! [K2: int,Nb: num] : aa_int_int(bit_se6526347334894502574or_int(K2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(K2),neg_numeral_sub_int(Nb,one))) ).

% xor_minus_numerals(2)
tff(fact_9656_sub__BitM__One__eq,axiom,
    ! [Nb: num] : neg_numeral_sub_int(bitM(Nb),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),neg_numeral_sub_int(Nb,one)) ).

% sub_BitM_One_eq
tff(fact_9657_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9658_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_aay(L),code_divmod_abs(K2,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9659_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bt,aTP_Lamm_bu) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9660_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_9661_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_aba),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_9662_Gcd__remove0__nat,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( gcd_Gcd_nat(M6) = gcd_Gcd_nat(minus_minus_set_nat(M6,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9663_Gcd__nat__eq__one,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(member_nat(one_one_nat),N3)
     => ( gcd_Gcd_nat(N3) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9664_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y3: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)) ).

% eq_Abs_Integ
tff(fact_9665_nat_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xb) ).

% nat.abs_eq
tff(fact_9666_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9667_int__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),zero_zero_nat)) ).

% int_def
tff(fact_9668_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_abb),Xb)) ).

% uminus_int.abs_eq
tff(fact_9669_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9670_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_abd),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_9671_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_abf),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_9672_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_abh),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_9673_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_abj),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_9674_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),inc(aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_9675_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q3)) = Q3 ).

% num_of_nat_numeral_eq
tff(fact_9676_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9677_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9678_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_9679_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_9680_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9681_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_abf),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9682_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_abd),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_9683_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_abk) ).

% prod_encode_def
tff(fact_9684_prod__encode__eq,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(nat_prod_encode,Xb) = aa_Pro4243173770044397973at_nat(nat_prod_encode,Y) )
    <=> ( Xb = Y ) ) ).

% prod_encode_eq
tff(fact_9685_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_1
tff(fact_9686_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_2
tff(fact_9687_nat_Orep__eq,axiom,
    ! [Xb: int] : aa_int_nat(nat2,Xb) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_9688_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K2,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K2)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_9689_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_abb)) ).

% uminus_int_def
tff(fact_9690_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9691_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_aba)) ).

% times_int_def
tff(fact_9692_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_abj)) ).

% minus_int_def
tff(fact_9693_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_abh)) ).

% plus_int_def
tff(fact_9694_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9695_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,sqr(aa_num_num(aa_num_fun_num_num(pow,Xb),Y))),Xb) ).

% pow.simps(3)
tff(fact_9696_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9697_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : sqr(aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,sqr(Nb))) ).

% sqr.simps(2)
tff(fact_9698_sqr_Osimps_I1_J,axiom,
    sqr(one) = one ).

% sqr.simps(1)
tff(fact_9699_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9700_sqr__conv__mult,axiom,
    ! [Xb: num] : sqr(Xb) = aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Xb) ).

% sqr_conv_mult
tff(fact_9701_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),aa_num_num(bit0,Y)) = sqr(aa_num_num(aa_num_fun_num_num(pow,Xb),Y)) ).

% pow.simps(2)
tff(fact_9702_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : sqr(aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,sqr(Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_9703_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_abl(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9704_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_9705_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_9706_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
              ( ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X3),n) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X7)
                    & ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_abl(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9707_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X3: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                            & ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_abl(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9708_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X3: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X3: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_abl(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9709_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_abl(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9710_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_abl(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9711_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_abl(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9712_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_abm(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9713_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,bitM(Nb))) ).

% and_minus_numerals(3)
tff(fact_9714_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).

% take_bit_num_simps(1)
tff(fact_9715_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9716_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9717_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abn,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_9718_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_9719_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abn,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_9720_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb))) ).

% take_bit_num_simps(7)
tff(fact_9721_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_9722_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_9723_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,bitM(Nb))) ).

% and_minus_numerals(7)
tff(fact_9724_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9725_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_abo(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9726_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_abp,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9727_and__not__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(4)
tff(fact_9728_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9729_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_9730_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9731_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_9732_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_9733_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_abq(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9734_and__not__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(7)
tff(fact_9735_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q3: num] :
      ( ( bit_and_not_num(Mb,Nb) = some_num(Q3) )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9736_and__not__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_abr,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_9737_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9738_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_9739_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_9740_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_9741_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_abv,product_Pair_nat_num(Nb,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9742_and__not__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( bit_and_not_num(Xb,Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != none_num ) ) )
       => ( ( ( Xb = one )
           => ( ? [N: num] : Xaa = aa_num_num(bit0,N)
             => ( Y != some_num(one) ) ) )
         => ( ( ( Xb = one )
             => ( ? [N: num] : Xaa = aa_num_num(bit1,N)
               => ( Y != none_num ) ) )
           => ( ! [M: num] :
                  ( ( Xb = aa_num_num(bit0,M) )
                 => ( ( Xaa = one )
                   => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M,N)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M,N)) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa_num_num(bit1,M) )
                       => ( ( Xaa = one )
                         => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_abr,bit_and_not_num(M,N)) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9743_and__not__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(5)
tff(fact_9744_and__not__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(6)
tff(fact_9745_and__not__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(9)
tff(fact_9746_and__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( Xb = one )
           => ( ? [N: num] : Xaa = aa_num_num(bit0,N)
             => ( Y != none_num ) ) )
         => ( ( ( Xb = one )
             => ( ? [N: num] : Xaa = aa_num_num(bit1,N)
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M: num] : Xb = aa_num_num(bit0,M)
               => ( ( Xaa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ) ) )
                 => ( ( ? [M: num] : Xb = aa_num_num(bit1,M)
                     => ( ( Xaa = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_abr,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9747_xor__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != none_num ) ) )
       => ( ( ( Xb = one )
           => ! [N: num] :
                ( ( Xaa = aa_num_num(bit0,N) )
               => ( Y != some_num(aa_num_num(bit1,N)) ) ) )
         => ( ( ( Xb = one )
             => ! [N: num] :
                  ( ( Xaa = aa_num_num(bit1,N) )
                 => ( Y != some_num(aa_num_num(bit0,N)) ) ) )
           => ( ! [M: num] :
                  ( ( Xb = aa_num_num(bit0,M) )
                 => ( ( Xaa = one )
                   => ( Y != some_num(aa_num_num(bit1,M)) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa_num_num(bit0,M) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa_num_num(bit0,M) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa_num_num(bit1,M) )
                       => ( ( Xaa = one )
                         => ( Y != some_num(aa_num_num(bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa_num_num(bit1,M) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa_num_num(bit1,M) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9748_and__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),one) = some_num(one) ).

% and_num.simps(1)
tff(fact_9749_xor__num_Osimps_I1_J,axiom,
    aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),one) = none_num ).

% xor_num.simps(1)
tff(fact_9750_xor__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(5)
tff(fact_9751_and__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(5)
tff(fact_9752_and__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,Nb)) = some_num(one) ).

% and_num.simps(3)
tff(fact_9753_and__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_9754_and__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,Nb)) = none_num ).

% and_num.simps(2)
tff(fact_9755_and__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),one) = none_num ).

% and_num.simps(4)
tff(fact_9756_and__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(8)
tff(fact_9757_and__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(6)
tff(fact_9758_xor__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(9)
tff(fact_9759_xor__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,Nb)) = some_num(aa_num_num(bit1,Nb)) ).

% xor_num.simps(2)
tff(fact_9760_xor__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,Nb)) = some_num(aa_num_num(bit0,Nb)) ).

% xor_num.simps(3)
tff(fact_9761_xor__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),one) = some_num(aa_num_num(bit1,Mb)) ).

% xor_num.simps(4)
tff(fact_9762_xor__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),one) = some_num(aa_num_num(bit0,Mb)) ).

% xor_num.simps(7)
tff(fact_9763_and__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_abr,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(9)
tff(fact_9764_xor__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(8)
tff(fact_9765_xor__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(6)
tff(fact_9766_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_9767_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_9768_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_abw) ).

% Rats_eq_int_div_nat
tff(fact_9769_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(A),B))),aa_int_rat(fract(A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),B)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9770_Rats__abs__iff,axiom,
    ! [Xb: real] :
      ( member_real(aa_real_real(abs_abs_real,Xb),field_5140801741446780682s_real)
    <=> member_real(Xb,field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_9771_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ).

% mult_rat
tff(fact_9772_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : divide_divide_rat(aa_int_rat(fract(A),B),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% divide_rat
tff(fact_9773_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ).

% less_rat
tff(fact_9774_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% add_rat
tff(fact_9775_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ).

% le_rat
tff(fact_9776_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( minus_minus_rat(aa_int_rat(fract(A),B),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% diff_rat
tff(fact_9777_sgn__rat,axiom,
    ! [A: int,B: int] : sgn_sgn_rat(aa_int_rat(fract(A),B)) = aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B))) ).

% sgn_rat
tff(fact_9778_Rats__no__bot__less,axiom,
    ! [Xb: real] :
    ? [X4: real] :
      ( member_real(X4,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Xb) ) ).

% Rats_no_bot_less
tff(fact_9779_Rats__dense__in__real,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [X4: real] :
          ( member_real(X4,field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X4)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y) ) ) ).

% Rats_dense_in_real
tff(fact_9780_Rats__no__top__le,axiom,
    ! [Xb: real] :
    ? [X4: real] :
      ( member_real(X4,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),X4) ) ).

% Rats_no_top_le
tff(fact_9781_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(fract(A),B) = aa_int_rat(fract(C),D) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9782_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_rat(fract(A),B) ) ) ).

% mult_rat_cancel
tff(fact_9783_eq__rat_I2_J,axiom,
    ! [A: int] : aa_int_rat(fract(A),zero_zero_int) = aa_int_rat(fract(zero_zero_int),one_one_int) ).

% eq_rat(2)
tff(fact_9784_Fract__of__nat__eq,axiom,
    ! [K2: nat] : aa_int_rat(fract(aa_nat_int(semiri1314217659103216013at_int,K2)),one_one_int) = aa_nat_rat(semiri681578069525770553at_rat,K2) ).

% Fract_of_nat_eq
tff(fact_9785_One__rat__def,axiom,
    one_one_rat = aa_int_rat(fract(one_one_int),one_one_int) ).

% One_rat_def
tff(fact_9786_Fract__of__int__eq,axiom,
    ! [K2: int] : aa_int_rat(fract(K2),one_one_int) = aa_int_rat(ring_1_of_int_rat,K2) ).

% Fract_of_int_eq
tff(fact_9787_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(fract(zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_9788_rat__number__expand_I3_J,axiom,
    ! [K2: num] : numeral_numeral_rat(K2) = aa_int_rat(fract(aa_num_int(numeral_numeral_int,K2)),one_one_int) ).

% rat_number_expand(3)
tff(fact_9789_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_num_int(numeral_numeral_int,W)),one_one_int) = numeral_numeral_rat(W) ).

% rat_number_collapse(3)
tff(fact_9790_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_9791_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% Fract_less_one_iff
tff(fact_9792_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_abx) ).

% Rats_eq_int_div_int
tff(fact_9793_rat__number__collapse_I5_J,axiom,
    aa_int_rat(fract(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = uminus_uminus_rat(one_one_rat) ).

% rat_number_collapse(5)
tff(fact_9794_Fract__add__one,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,Mb),Nb)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(fract(Mb),Nb)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9795_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_9796_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% Fract_le_one_iff
tff(fact_9797_rat__number__expand_I5_J,axiom,
    ! [K2: num] : uminus_uminus_rat(numeral_numeral_rat(K2)) = aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int) ).

% rat_number_expand(5)
tff(fact_9798_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = uminus_uminus_rat(numeral_numeral_rat(W)) ).

% rat_number_collapse(4)
tff(fact_9799_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aby,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xb),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_9800_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9801_bij__betw__Suc,axiom,
    ! [M6: set_nat,N3: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N3)
    <=> ( image_nat_nat(suc,M6) = N3 ) ) ).

% bij_betw_Suc
tff(fact_9802_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9803_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9804_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A4)) ).

% zero_notin_Suc_image
tff(fact_9805_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_abd)) ).

% less_int_def
tff(fact_9806_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_abf)) ).

% less_eq_int_def
tff(fact_9807_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9808_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb) ).

% image_Suc_lessThan
tff(fact_9809_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9810_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9811_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9812_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9813_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9814_positive__rat,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(positive,aa_int_rat(fract(A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% positive_rat
tff(fact_9815_Rat_Opositive__mult,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(positive,Xb)
     => ( aa_rat_o(positive,Y)
       => aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y)) ) ) ).

% Rat.positive_mult
tff(fact_9816_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,minus_minus_rat(Y,Xb)) ) ).

% less_rat_def
tff(fact_9817_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_abz(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_9818_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(aa_rat7754553326175492643nt_int(rep_Rat,Xb))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,Xb)))) ) ).

% Rat.positive.rep_eq
tff(fact_9819_Rat_Opositive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_aca) ).

% Rat.positive_def
tff(fact_9820_of__rat__dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [Q2: rat] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),field_7254667332652039916t_real(Q2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,field_7254667332652039916t_real(Q2)),Y) ) ) ).

% of_rat_dense
tff(fact_9821_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_acb) ).

% plus_rat_def
tff(fact_9822_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_acc) ).

% inverse_rat_def
tff(fact_9823_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat_def
tff(fact_9824_Fract_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      aa_int_rat(fract(Xaa),Xb) = aa_Pro6166618844887768661nt_rat(abs_Rat,
        $ite(Xb = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xaa),Xb))) ).

% Fract.abs_eq
tff(fact_9825_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat_def
tff(fact_9826_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_acd) ).

% times_rat_def
tff(fact_9827_plus__rat_Oabs__eq,axiom,
    ! [Xaa: product_prod_int_int,Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(ratrel(Xaa),Xaa)
     => ( aa_Pro4592068669816069595_int_o(ratrel(Xb),Xb)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xaa)),aa_Pro6166618844887768661nt_rat(abs_Rat,Xb)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xaa)),product_snd_int_int(Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xb)),product_snd_int_int(Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xaa)),product_snd_int_int(Xb)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_9828_inverse__rat_Oabs__eq,axiom,
    ! [Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(ratrel(Xb),Xb)
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xb)) = aa_Pro6166618844887768661nt_rat(abs_Rat,
            $ite(product_fst_int_int(Xb) = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(Xb)),product_fst_int_int(Xb)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_9829_Rat_Opositive_Oabs__eq,axiom,
    ! [Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(ratrel(Xb),Xb)
     => ( aa_rat_o(positive,aa_Pro6166618844887768661nt_rat(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_9830_ratrel__iff,axiom,
    ! [Xb: product_prod_int_int,Y: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(ratrel(Xb),Y)
    <=> ( ( product_snd_int_int(Xb) != zero_zero_int )
        & ( product_snd_int_int(Y) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xb)),product_snd_int_int(Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Y)),product_snd_int_int(Xb)) ) ) ) ).

% ratrel_iff
tff(fact_9831_one__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(ratrel(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) ).

% one_rat.rsp
tff(fact_9832_zero__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(ratrel(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) ).

% zero_rat.rsp
tff(fact_9833_ratrel__def,axiom,
    ! [X: product_prod_int_int,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(ratrel(X),Xa)
    <=> ( ( product_snd_int_int(X) != zero_zero_int )
        & ( product_snd_int_int(Xa) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Xa)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa)),product_snd_int_int(X)) ) ) ) ).

% ratrel_def
tff(fact_9834_times__rat_Oabs__eq,axiom,
    ! [Xaa: product_prod_int_int,Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(ratrel(Xaa),Xaa)
     => ( aa_Pro4592068669816069595_int_o(ratrel(Xb),Xb)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xaa)),aa_Pro6166618844887768661nt_rat(abs_Rat,Xb)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xaa)),product_fst_int_int(Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xaa)),product_snd_int_int(Xb)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_9835_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).

% min_Suc_Suc
tff(fact_9836_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9837_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_9838_min__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K2))) ).

% min_Suc_numeral
tff(fact_9839_min__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K2)),Nb)) ).

% min_numeral_Suc
tff(fact_9840_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)) ).

% nat_mult_min_right
tff(fact_9841_nat__mult__min__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_min_left
tff(fact_9842_min__diff,axiom,
    ! [Mb: nat,I: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),I) ).

% min_diff
tff(fact_9843_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9844_concat__bit__assoc__sym,axiom,
    ! [Mb: nat,Nb: nat,K2: int,L: int,R2: int] : aa_int_int(bit_concat_bit(Mb,aa_int_int(bit_concat_bit(Nb,K2),L)),R2) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb),K2),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_9845_take__bit__concat__bit__eq,axiom,
    ! [Mb: nat,Nb: nat,K2: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_concat_bit(Nb,K2),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb),K2),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9846_min__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ace,Nb)),Mb) ).

% min_Suc1
tff(fact_9847_min__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_acf,Nb)),Mb) ).

% min_Suc2
tff(fact_9848_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q3) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9849_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q3),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9850_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_9851_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(S3)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S3))) ) ).

% card_le_Suc_Max
tff(fact_9852_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_acg(Mb),Nb)))) ).

% divide_nat_def
tff(fact_9853_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ach(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9854_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9855_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9856_list__encode_Ocases,axiom,
    ! [Xb: list_nat] :
      ( ( Xb != nil_nat )
     => ~ ! [X4: nat,Xs2: list_nat] : Xb != cons_nat(X4,Xs2) ) ).

% list_encode.cases
tff(fact_9857_list__encode_Oelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xb) = Y )
     => ( ( ( Xb = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( Xb = cons_nat(X4,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9858_Inf__real__def,axiom,
    ! [X6: set_real] : comple4887499456419720421f_real(X6) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X6))) ).

% Inf_real_def
tff(fact_9859_list__encode__eq,axiom,
    ! [Xb: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xb) = aa_list_nat_nat(nat_list_encode,Y) )
    <=> ( Xb = Y ) ) ).

% list_encode_eq
tff(fact_9860_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9861_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_9862_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K2)),cons_nat(K2,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9863_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K2)),cons_nat(aa_nat_nat(suc,K2),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9864_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aci,Nb),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ) ) ).

% range_mod
tff(fact_9865_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_9866_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9867_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9868_range__mult,axiom,
    ! [A: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9869_surj__list__encode,axiom,
    image_list_nat_nat(nat_list_encode,top_top_set_list_nat) = top_top_set_nat ).

% surj_list_encode
tff(fact_9870_bij__list__encode,axiom,
    aa_set_nat_o(bij_be8532844293280997160at_nat(nat_list_encode,top_top_set_list_nat),top_top_set_nat) ).

% bij_list_encode
tff(fact_9871_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_9872_bij__prod__encode,axiom,
    aa_set_nat_o(bij_be5333170631980326235at_nat(nat_prod_encode,top_to4669805908274784177at_nat),top_top_set_nat) ).

% bij_prod_encode
tff(fact_9873_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_acj,Nb)),Xb)) ).

% root_def
tff(fact_9874_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_9875_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9876_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : size_size_char(char2((X1),(X2),(X32),(X42),(X52),(X62),(X72),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_9877_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_9878_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9879_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B72))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B62)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B52)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B42)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B32)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B22)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B1)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_9880_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_9881_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_9882_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_9883_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_9884_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_9885_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_9886_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_9887_nth__upto,axiom,
    ! [I: int,K2: 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,K2))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K2)) ) ) ).

% nth_upto
tff(fact_9888_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9889_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9890_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9891_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9892_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9893_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_9894_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_9895_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).

% distinct_upto
tff(fact_9896_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_9897_upto__split2,axiom,
    ! [I: int,J: int,K2: 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),K2)
       => ( upto(I,K2) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K2)) ) ) ) ).

% upto_split2
tff(fact_9898_upto__split1,axiom,
    ! [I: int,J: int,K2: 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),K2)
       => ( upto(I,K2) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K2)) ) ) ) ).

% upto_split1
tff(fact_9899_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9900_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9901_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_9902_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_9903_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_9904_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_9905_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9906_upto__split3,axiom,
    ! [I: int,J: int,K2: 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),K2)
       => ( upto(I,K2) = 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),K2))) ) ) ) ).

% upto_split3
tff(fact_9907_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9908_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xb != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D3 = 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,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D3,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9909_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9910_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => 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_9911_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => 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_9912_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F3,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F3,Z3)) ) ) ) ) ).

% MVT2
tff(fact_9913_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)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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),D2)
                 => 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_9914_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)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [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),D2)
                 => 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_9915_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(F,K2,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),K2) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9916_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(F,K2,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) = K2 ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9917_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_9918_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(V,K2,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9919_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_9920_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_9921_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_9922_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ack,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S2)) ).

% DERIV_pow
tff(fact_9923_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_acl(G),Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9924_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_acm,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_9925_DERIV__log,axiom,
    ! [Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9926_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_acn(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))))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9927_DERIV__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xb,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_aco(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,Mb),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9928_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))),numeral_numeral_real(aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9929_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F3: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_acp(F),N),aa_nat_real(aa_real_fun_nat_real(F3,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X4: real] :
            ( member_real(X4,set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X4)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F3,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X4: real,Y3: real] :
                    ( member_real(X4,set_or1633881224788618240n_real(A,B))
                   => ( member_real(Y3,set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X4),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,X4),Y3)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_acq(F),suminf_real(aa_real_fun_nat_real(F3,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9930_DERIV__arctan,axiom,
    ! [Xb: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9931_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A4)) ).

% arsinh_real_has_field_derivative
tff(fact_9932_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D3: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D3 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
           => ( D3 = 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)))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D3,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9933_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: 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,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9934_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: 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,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_9935_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X4: real] :
          ( member_real(X4,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_acr(F),X4)) )
     => ( 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_act(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_acr(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9936_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9937_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_9938_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_9939_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9940_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9941_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9942_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T3: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [T3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_no(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9943_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ? [T3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_no(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9944_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T3: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [T3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_no(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9945_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xb != zero_zero_real )
         => ( ! [M: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T3))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9946_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,T3: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_acu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9947_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
                 => ( ( Xb != C )
                   => ? [T3: 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),T3)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T3)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),Xb) ) )
                        & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acv(Diff),C),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9948_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acw(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9949_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),C)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acw(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9950_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K2: nat,B4: real] :
      ( ! [M: nat,T3: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K2) )
       => ! [M2: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_acy(Nb),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)),T4)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_acz(Diff),M2),T4),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,T4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T4,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9951_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_ada,suminf_real(aa_real_fun_nat_real(aTP_Lamm_adb,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9952_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)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X: real] :
                ( ( ( X != 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),X))),R4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9953_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 )
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X: real] :
                ( ( ( X != 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),X))),R4) )
               => ( aa_real_real(F,X) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9954_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)
       => ? [R4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
            & ! [X: real] :
                ( ( ( X != 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),X))),R4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9955_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_9956_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_adc,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9957_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_9958_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_9959_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_9960_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_9961_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
                 => 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,A),Z3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F3,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F3,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9962_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_add(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9963_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_add(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))),groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9964_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_9965_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_ai,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9966_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_ade,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9967_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_gd(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9968_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R4: real] :
        ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R4),aa_nat_real(X6,N)) )
     => filterlim_nat_real(aTP_Lamm_adf(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9969_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_adg,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9970_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_adh,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9971_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_adi,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9972_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adj,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9973_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)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N7)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9974_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_9975_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adk(Xb),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9976_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_9977_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_9978_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_adl,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9979_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adm,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9980_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adn,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9981_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ado,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9982_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_add(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9983_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_add(A)) ) ) ) ).

% summable
tff(fact_9984_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adp(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_adq(Theta),K),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9985_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_adr(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_adq(Theta),K),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9986_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_ads(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_add(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9987_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_jy,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9988_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),suminf_real(aTP_Lamm_add(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9989_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_ads(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_add(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9990_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))),L3)
              & filterlim_nat_real(aTP_Lamm_ads(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_adt(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9991_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_adt(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_add(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9992_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_add(A))),groups6591440286371151544t_real(aTP_Lamm_add(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9993_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_adt(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_add(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9994_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_adu(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9995_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_adv(P),K2),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9996_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F4) ) ).

% le_sequentially
tff(fact_9997_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
         => aa_nat_o(P,X4) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9998_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_9999_sequentially__offset,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_adv(P),K2),at_top_nat) ) ).

% sequentially_offset
tff(fact_10000_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_10001_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C3: real] :
        ! [X3: real] : aa_real_real(F,X3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),C3) ) ).

% real_bounded_linear
tff(fact_10002_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_10003_artanh__real__at__left__1,axiom,
    filterlim_real_real(artanh_real,at_top_real,topolo2177554685111907308n_real(one_one_real,set_or5984915006950818249n_real(one_one_real))) ).

% artanh_real_at_left_1
tff(fact_10004_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_adw,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10005_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10006_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10007_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adx(Nb),F),at_top_real,F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10008_dist__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V3694042436643373181omplex(Xb),Y) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y)) ).

% dist_complex_def
tff(fact_10009_dist__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Xb),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).

% dist_real_def
tff(fact_10010_tanh__real__at__bot,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,one_one_real)),at_bot_real) ).

% tanh_real_at_bot
tff(fact_10011_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adx(Nb),F),at_bot_real,F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10012_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10013_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ady,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10014_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_10015_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real(arcosh_real,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(one_one_real,set_or5849166863359141190n_real(one_one_real))) ).

% tendsto_arcosh_at_left_1
tff(fact_10016_artanh__real__at__right__1,axiom,
    filterlim_real_real(artanh_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,one_one_real),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,one_one_real)))) ).

% artanh_real_at_right_1
tff(fact_10017_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K2) ).

% atLeast_Suc_greaterThan
tff(fact_10018_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_10019_greaterThan__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K2)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K2),insert_nat(aa_nat_nat(suc,K2),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_10020_atLeast__Suc,axiom,
    ! [K2: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K2)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K2),insert_nat(K2,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_10021_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X4: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ( ! [X4: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
         => ( ! [X4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
           => ( ! [X4: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X4,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_10022_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ? [L3: real,Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z3,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_10023_continuous__on__arcosh,axiom,
    ! [A4: set_real] :
      ( ord_less_eq_set_real(A4,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A4,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_10024_continuous__on__arcosh_H,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A4,F)
     => ( ! [X4: real] :
            ( member_real(X4,A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X4)) )
       => topolo5044208981011980120l_real(A4,aTP_Lamm_adz(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_10025_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).

% continuous_on_arccos'
tff(fact_10026_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).

% continuous_on_arcsin'
tff(fact_10027_continuous__on__artanh,axiom,
    ! [A4: set_real] :
      ( ord_less_eq_set_real(A4,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A4,artanh_real) ) ).

% continuous_on_artanh
tff(fact_10028_continuous__on__artanh_H,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A4,F)
     => ( ! [X4: real] :
            ( member_real(X4,A4)
           => member_real(aa_real_real(F,X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A4,aTP_Lamm_aea(F)) ) ) ).

% continuous_on_artanh'
tff(fact_10029_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi),B)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F3,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_10030_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10031_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_10032_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_10033_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aeb,K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_10034_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_acj,Nb),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_10035_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => inj_on_real_real(aa_rea7340605660300375871l_real(log,B),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_10036_inj__list__encode,axiom,
    ! [A4: set_list_nat] : inj_on_list_nat_nat(nat_list_encode,A4) ).

% inj_list_encode
tff(fact_10037_inj__Suc,axiom,
    ! [N3: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N3) ).

% inj_Suc
tff(fact_10038_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : inj_on2178005380612969504at_nat(nat_prod_encode,A4) ).

% inj_prod_encode
tff(fact_10039_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K2: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aby,K2)),N3) ) ).

% inj_on_diff_nat
tff(fact_10040_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_10041_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% inj_on_char_of_nat
tff(fact_10042_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_10043_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_10044_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_10045_list__encode_Opelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,Xb) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),Xb)
       => ( ( ( Xb = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( Xb = cons_nat(X4,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X4,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_10046_remdups__upt,axiom,
    ! [Mb: nat,Nb: nat] : remdups_nat(upt(Mb,Nb)) = upt(Mb,Nb) ).

% remdups_upt
tff(fact_10047_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_10048_drop__upt,axiom,
    ! [Mb: nat,I: nat,J: nat] : drop_nat(Mb,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb),J) ).

% drop_upt
tff(fact_10049_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_10050_take__upt,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)),Nb)
     => ( take_nat(Mb,upt(I,Nb)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)) ) ) ).

% take_upt
tff(fact_10051_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_10052_sorted__list__of__set__range,axiom,
    ! [Mb: nat,Nb: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = upt(Mb,Nb) ).

% sorted_list_of_set_range
tff(fact_10053_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_10054_nth__upt,axiom,
    ! [I: nat,K2: 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),K2)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) ) ) ).

% nth_upt
tff(fact_10055_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Mb),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_10056_atMost__upto,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_10057_atLeast__upt,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_10058_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_10059_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list_nat,Q3: nat] :
      ( ( cons_nat(Mb,cons_nat(Nb,Ns)) = upt(Mb,Q3) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Mb),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_10060_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_10061_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_10062_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I),J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_10063_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb) = set_nat2(upt(Nb,aa_nat_nat(suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_10064_distinct__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I,J)) ).

% distinct_upt
tff(fact_10065_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(suc,upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_10066_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aec,Nb),upt(zero_zero_nat,Mb)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% map_add_upt
tff(fact_10067_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_10068_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(aTP_Lamm_aed,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_10069_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K2: 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),K2)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% upt_add_eq_append
tff(fact_10070_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_10071_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_10072_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_10073_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_10074_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : tl_nat(upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),Nb) ).

% tl_upt
tff(fact_10075_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups4561878855575611511st_nat(upt(Mb,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_an),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) ) ) ).

% sum_list_upt
tff(fact_10076_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aee(Mb),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_aef(Mb),N3)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aeg(Mb),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_10077_card__length__sum__list,axiom,
    ! [Mb: nat,N3: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aee(Mb),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),Mb)),one_one_nat)),N3) ).

% card_length_sum_list
tff(fact_10078_sorted__upto,axiom,
    ! [Mb: int,Nb: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Mb,Nb)) ).

% sorted_upto
tff(fact_10079_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_10080_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_10081_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_10082_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_10083_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aeh)) ).

% pred_nat_def
tff(fact_10084_less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% less_eq
tff(fact_10085_pred__nat__trancl__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% pred_nat_trancl_eq_le
tff(fact_10086_product__atMost__eq__Un,axiom,
    ! [A4: set_nat,Mb: nat] : produc457027306803732586at_nat(A4,aTP_Lamm_aei(Mb)) = sup_su6327502436637775413at_nat(produc457027306803732586at_nat(A4,aTP_Lamm_aej(Mb)),produc457027306803732586at_nat(A4,aTP_Lamm_aek(Mb))) ).

% product_atMost_eq_Un
tff(fact_10087_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_sy(Mb))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,Mb),aTP_Lamm_aej(Mb)) ).

% pairs_le_eq_Sigma
tff(fact_10088_coprime__Suc__right__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,Nb)) ).

% coprime_Suc_right_nat
tff(fact_10089_coprime__Suc__left__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,Nb)),Nb) ).

% coprime_Suc_left_nat
tff(fact_10090_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,B),C)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D) )
        <=> ( ( A = B )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_10091_coprime__Suc__0__right,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_10092_coprime__Suc__0__left,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ).

% coprime_Suc_0_left
tff(fact_10093_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),B)
         => ( Xb = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_10094_coprime__diff__one__left__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),Nb) ) ).

% coprime_diff_one_left_nat
tff(fact_10095_coprime__diff__one__right__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_10096_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( member_real(Xb,field_5140801741446780682s_real)
     => ~ ! [M: 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,M)),aa_nat_real(semiri5074537144036343181t_real,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M),N) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_10097_coprime__abs__left__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_int_int(abs_abs_int,K2)),L)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K2),L) ) ).

% coprime_abs_left_iff
tff(fact_10098_coprime__abs__right__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K2),aa_int_int(abs_abs_int,L))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K2),L) ) ).

% coprime_abs_right_iff
tff(fact_10099_coprime__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Mb),Nb) ) ).

% coprime_int_iff
tff(fact_10100_coprime__nat__abs__left__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),Nb)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K2),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% coprime_nat_abs_left_iff
tff(fact_10101_coprime__nat__abs__right__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_int_nat(nat2,aa_int_int(abs_abs_int,K2)))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K2) ) ).

% coprime_nat_abs_right_iff
tff(fact_10102_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),B)
         => ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_10103_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,B),C)
       => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,B)),aa_int_int(abs_abs_int,D)) )
        <=> ( ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) )
            & ( aa_int_int(abs_abs_int,C) = aa_int_int(abs_abs_int,D) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_10104_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A4: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Nb),A4)) = insert_nat(Nb,vimage_nat_nat(suc,A4)) ).

% vimage_Suc_insert_Suc
tff(fact_10105_vimage__Suc__insert__0,axiom,
    ! [A4: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A4)) = vimage_nat_nat(suc,A4) ).

% vimage_Suc_insert_0
tff(fact_10106_finite__vimage__Suc__iff,axiom,
    ! [F4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_nat(suc,F4))
    <=> aa_set_nat_o(finite_finite_nat,F4) ) ).

% finite_vimage_Suc_iff
tff(fact_10107_set__decode__div__2,axiom,
    ! [Xb: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,Xb)) ).

% set_decode_div_2
tff(fact_10108_set__encode__vimage__Suc,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A4)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_10109_Field__natLeq__on,axiom,
    ! [Nb: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ael(Nb)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bu,Nb)) ).

% Field_natLeq_on
tff(fact_10110_wf__less,axiom,
    wf_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_10111_ATP_Olambda__1,axiom,
    ! [Uu2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(aTP_Lamm_acc,Uu2) = $ite(product_fst_int_int(Uu2) = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,product_snd_int_int(Uu2)),product_fst_int_int(Uu2))) ).

% ATP.lambda_1
tff(fact_10112_ATP_Olambda__2,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_mk,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu2)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_2
tff(fact_10113_ATP_Olambda__3,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_nc,Uu2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_3
tff(fact_10114_ATP_Olambda__4,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_ze,Uu2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu2),numeral_numeral_real(aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).

% ATP.lambda_4
tff(fact_10115_ATP_Olambda__5,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_mj,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu2)) ).

% ATP.lambda_5
tff(fact_10116_ATP_Olambda__6,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_adc,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu2)),aa_real_real(sin_real,Uu2)) ).

% ATP.lambda_6
tff(fact_10117_ATP_Olambda__7,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_ade,Uu2) = aa_real_real(aa_nat_fun_real_real(root,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_7
tff(fact_10118_ATP_Olambda__8,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_aed,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_8
tff(fact_10119_ATP_Olambda__9,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_nw,Uu2) = aa_complex_complex(plus_plus_complex(Uu2),one_one_complex) ).

% ATP.lambda_9
tff(fact_10120_ATP_Olambda__10,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_ny,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),one_one_real) ).

% ATP.lambda_10
tff(fact_10121_ATP_Olambda__11,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_oa,Uu2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),one_one_rat) ).

% ATP.lambda_11
tff(fact_10122_ATP_Olambda__12,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_nz,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),one_one_nat) ).

% ATP.lambda_12
tff(fact_10123_ATP_Olambda__13,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_nx,Uu2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),one_one_int) ).

% ATP.lambda_13
tff(fact_10124_ATP_Olambda__14,axiom,
    ! [Uu2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aca,Uu2)
    <=> 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(Uu2)),product_snd_int_int(Uu2))) ) ).

% ATP.lambda_14
tff(fact_10125_ATP_Olambda__15,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_adg,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_15
tff(fact_10126_ATP_Olambda__16,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_ada,Uu2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_kc,Uu2)) ).

% ATP.lambda_16
tff(fact_10127_ATP_Olambda__17,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_adi,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ).

% ATP.lambda_17
tff(fact_10128_ATP_Olambda__18,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_abr,Uu2) = some_num(aa_num_num(bit1,Uu2)) ).

% ATP.lambda_18
tff(fact_10129_ATP_Olambda__19,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_abn,Uu2) = some_num(aa_num_num(bit0,Uu2)) ).

% ATP.lambda_19
tff(fact_10130_ATP_Olambda__20,axiom,
    ! [Uu2: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_jl,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_20
tff(fact_10131_ATP_Olambda__21,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_abw,Uu2)
    <=> ? [I3: int,N4: nat] :
          ( ( Uu2 = 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_21
tff(fact_10132_ATP_Olambda__22,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_abx,Uu2)
    <=> ? [I3: int,J3: int] :
          ( ( Uu2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_22
tff(fact_10133_ATP_Olambda__23,axiom,
    ! [Uu2: nat] : aa_nat_option_num(aTP_Lamm_abp,Uu2) = some_num(one) ).

% ATP.lambda_23
tff(fact_10134_ATP_Olambda__24,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abu(Uu2),Uua) = case_num_option_num(some_num(one),aTP_Lamm_abs(Uua),aTP_Lamm_abt(Uua),Uu2) ).

% ATP.lambda_24
tff(fact_10135_ATP_Olambda__25,axiom,
    ! [Uu2: complex,Uua: nat] :
      aa_nat_complex(aTP_Lamm_zc(Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua)),zero_zero_complex) ).

% ATP.lambda_25
tff(fact_10136_ATP_Olambda__26,axiom,
    ! [Uu2: real,Uua: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zb,Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)),zero_zero_real) ).

% ATP.lambda_26
tff(fact_10137_ATP_Olambda__27,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_wd(Uu2),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,Uu2),Uua)),zero_zero_complex) ).

% ATP.lambda_27
tff(fact_10138_ATP_Olambda__28,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wg,Uu2),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,Uu2),Uua)),zero_zero_real) ).

% ATP.lambda_28
tff(fact_10139_ATP_Olambda__29,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_wf(Uu2),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,Uu2),Uua)),zero_zero_rat) ).

% ATP.lambda_29
tff(fact_10140_ATP_Olambda__30,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_we,Uu2),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,Uu2),Uua)),zero_zero_int) ).

% ATP.lambda_30
tff(fact_10141_ATP_Olambda__31,axiom,
    ! [Uu2: complex,Uua: nat] :
      aa_nat_complex(aTP_Lamm_za(Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_complex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua))) ).

% ATP.lambda_31
tff(fact_10142_ATP_Olambda__32,axiom,
    ! [Uu2: real,Uua: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yz,Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua))) ).

% ATP.lambda_32
tff(fact_10143_ATP_Olambda__33,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_nd(Uu2),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(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_33
tff(fact_10144_ATP_Olambda__34,axiom,
    ! [Uu2: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_zz,Uu2),Uua) = $ite(Uu2 = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu2)),Uua)),aa_int_int(abs_abs_int,Uu2))) ).

% ATP.lambda_34
tff(fact_10145_ATP_Olambda__35,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_wh(Uu2),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,Uu2),Uua)),zero_zero_complex) ).

% ATP.lambda_35
tff(fact_10146_ATP_Olambda__36,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wk,Uu2),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,Uu2),Uua)),zero_zero_real) ).

% ATP.lambda_36
tff(fact_10147_ATP_Olambda__37,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_wj(Uu2),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,Uu2),Uua)),zero_zero_rat) ).

% ATP.lambda_37
tff(fact_10148_ATP_Olambda__38,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wi,Uu2),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,Uu2),Uua)),zero_zero_int) ).

% ATP.lambda_38
tff(fact_10149_ATP_Olambda__39,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_abv,Uu2),Uua) = case_nat_option_num(none_num,aTP_Lamm_abu(Uua),Uu2) ).

% ATP.lambda_39
tff(fact_10150_ATP_Olambda__40,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_abs(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abn,bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_40
tff(fact_10151_ATP_Olambda__41,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abo(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_abn,bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_41
tff(fact_10152_ATP_Olambda__42,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ws(Uu2),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_wr(Uu2),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_42
tff(fact_10153_ATP_Olambda__43,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_wq(Uu2),Uua) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sz(Uu2),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_43
tff(fact_10154_ATP_Olambda__44,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tc(Uu2),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tb(Uu2),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_44
tff(fact_10155_ATP_Olambda__45,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ta(Uu2),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sz(Uu2),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_45
tff(fact_10156_ATP_Olambda__46,axiom,
    ! [Uu2: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_acb,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu2)),product_snd_int_int(Uua))) ).

% ATP.lambda_46
tff(fact_10157_ATP_Olambda__47,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kb,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_47
tff(fact_10158_ATP_Olambda__48,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nm,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_48
tff(fact_10159_ATP_Olambda__49,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nt,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_49
tff(fact_10160_ATP_Olambda__50,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xd(Uu2),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_50
tff(fact_10161_ATP_Olambda__51,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xc(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_51
tff(fact_10162_ATP_Olambda__52,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xg,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_52
tff(fact_10163_ATP_Olambda__53,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xf(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_53
tff(fact_10164_ATP_Olambda__54,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xe,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_54
tff(fact_10165_ATP_Olambda__55,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_vh(Uu2),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uua)),aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_55
tff(fact_10166_ATP_Olambda__56,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_56
tff(fact_10167_ATP_Olambda__57,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_vi(Uu2),Uua) = divide_divide_rat(aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uua),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_57
tff(fact_10168_ATP_Olambda__58,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zf,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_58
tff(fact_10169_ATP_Olambda__59,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zd,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_59
tff(fact_10170_ATP_Olambda__60,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_aam,Uu2),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu2)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_60
tff(fact_10171_ATP_Olambda__61,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_jt,Uu2),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(plus_plus_complex(m3),one_one_complex)) ) ).

% ATP.lambda_61
tff(fact_10172_ATP_Olambda__62,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jv,Uu2),Uua) = $let(
        m3: real,
        m3:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m3),one_one_real)) ) ).

% ATP.lambda_62
tff(fact_10173_ATP_Olambda__63,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_jw,Uu2),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m3),one_one_rat)) ) ).

% ATP.lambda_63
tff(fact_10174_ATP_Olambda__64,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jx,Uu2),Uua) = $let(
        m3: nat,
        m3:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_64
tff(fact_10175_ATP_Olambda__65,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ju,Uu2),Uua) = $let(
        m3: int,
        m3:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu2)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_65
tff(fact_10176_ATP_Olambda__66,axiom,
    ! [Uu2: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_acd,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_fst_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu2)),product_snd_int_int(Uua))) ).

% ATP.lambda_66
tff(fact_10177_ATP_Olambda__67,axiom,
    ! [Uu2: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_zw(Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_67
tff(fact_10178_ATP_Olambda__68,axiom,
    ! [Uu2: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_zx(Uu2),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu2),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_68
tff(fact_10179_ATP_Olambda__69,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kc,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_69
tff(fact_10180_ATP_Olambda__70,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adb,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_70
tff(fact_10181_ATP_Olambda__71,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_add(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_71
tff(fact_10182_ATP_Olambda__72,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xj(Uu2),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_72
tff(fact_10183_ATP_Olambda__73,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_xi(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_73
tff(fact_10184_ATP_Olambda__74,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xm,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_74
tff(fact_10185_ATP_Olambda__75,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xl(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_75
tff(fact_10186_ATP_Olambda__76,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_xk,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua))) ).

% ATP.lambda_76
tff(fact_10187_ATP_Olambda__77,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_sv(Uu2),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_77
tff(fact_10188_ATP_Olambda__78,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sx,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_78
tff(fact_10189_ATP_Olambda__79,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sw(Uu2),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_79
tff(fact_10190_ATP_Olambda__80,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ob(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu2),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu2),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_80
tff(fact_10191_ATP_Olambda__81,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_od,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu2),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_81
tff(fact_10192_ATP_Olambda__82,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oc(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu2),Uua)),minus_minus_rat(divide_divide_rat(Uu2,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_82
tff(fact_10193_ATP_Olambda__83,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ug(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu2),Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)) ).

% ATP.lambda_83
tff(fact_10194_ATP_Olambda__84,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ui,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu2),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_84
tff(fact_10195_ATP_Olambda__85,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_uh(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(Uu2),Uua)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),Uua)) ).

% ATP.lambda_85
tff(fact_10196_ATP_Olambda__86,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xb,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_86
tff(fact_10197_ATP_Olambda__87,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nv,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_87
tff(fact_10198_ATP_Olambda__88,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aek(Uu2),Uua) = aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)),Uu2) ).

% ATP.lambda_88
tff(fact_10199_ATP_Olambda__89,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aeb,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_89
tff(fact_10200_ATP_Olambda__90,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wn,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uu2) ).

% ATP.lambda_90
tff(fact_10201_ATP_Olambda__91,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wm,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_91
tff(fact_10202_ATP_Olambda__92,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_cq,Uu2),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu2) = one_one_complex ) ) ).

% ATP.lambda_92
tff(fact_10203_ATP_Olambda__93,axiom,
    ! [Uu2: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_cp,Uu2),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) = one_one_real ) ) ).

% ATP.lambda_93
tff(fact_10204_ATP_Olambda__94,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adn,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_94
tff(fact_10205_ATP_Olambda__95,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ady,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_95
tff(fact_10206_ATP_Olambda__96,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_adw,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),Uua))),Uua) ).

% ATP.lambda_96
tff(fact_10207_ATP_Olambda__97,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jy,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_97
tff(fact_10208_ATP_Olambda__98,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rg(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_98
tff(fact_10209_ATP_Olambda__99,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rh(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_99
tff(fact_10210_ATP_Olambda__100,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ri(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_100
tff(fact_10211_ATP_Olambda__101,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_rj(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_101
tff(fact_10212_ATP_Olambda__102,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jh(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_102
tff(fact_10213_ATP_Olambda__103,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_je(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_103
tff(fact_10214_ATP_Olambda__104,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_jg(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_104
tff(fact_10215_ATP_Olambda__105,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jf(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_105
tff(fact_10216_ATP_Olambda__106,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_im(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,aa_nat_nat(suc,Uua))),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_106
tff(fact_10217_ATP_Olambda__107,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ik(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_107
tff(fact_10218_ATP_Olambda__108,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_il(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,aa_nat_nat(suc,Uua))),aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_108
tff(fact_10219_ATP_Olambda__109,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gy(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_109
tff(fact_10220_ATP_Olambda__110,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ha(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_110
tff(fact_10221_ATP_Olambda__111,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gz(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_111
tff(fact_10222_ATP_Olambda__112,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mb(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_112
tff(fact_10223_ATP_Olambda__113,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jd(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_113
tff(fact_10224_ATP_Olambda__114,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jb(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,Uua),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_114
tff(fact_10225_ATP_Olambda__115,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jc(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_115
tff(fact_10226_ATP_Olambda__116,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ix(Uu2),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_116
tff(fact_10227_ATP_Olambda__117,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ja(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_117
tff(fact_10228_ATP_Olambda__118,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_iy(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,Uua),aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_118
tff(fact_10229_ATP_Olambda__119,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_iz(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_119
tff(fact_10230_ATP_Olambda__120,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kz(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_120
tff(fact_10231_ATP_Olambda__121,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kx(Uu2),Uua) = minus_minus_rat(aa_nat_rat(Uu2,Uua),aa_nat_rat(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_121
tff(fact_10232_ATP_Olambda__122,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ky(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_122
tff(fact_10233_ATP_Olambda__123,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pw(Uu2),Uua) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu2,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_123
tff(fact_10234_ATP_Olambda__124,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pz(Uu2),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu2,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_124
tff(fact_10235_ATP_Olambda__125,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_si(Uu2),Uua) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu2,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_125
tff(fact_10236_ATP_Olambda__126,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_sg(Uu2),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu2,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_126
tff(fact_10237_ATP_Olambda__127,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gk(Uu2),Uua)
    <=> ( aa_complex_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_127
tff(fact_10238_ATP_Olambda__128,axiom,
    ! [Uu2: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gm(Uu2),Uua)
    <=> ( aa_complex_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_128
tff(fact_10239_ATP_Olambda__129,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_go(Uu2),Uua)
    <=> ( aa_complex_nat(Uu2,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_129
tff(fact_10240_ATP_Olambda__130,axiom,
    ! [Uu2: fun_complex_int,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gq(Uu2),Uua)
    <=> ( aa_complex_int(Uu2,Uua) = zero_zero_int ) ) ).

% ATP.lambda_130
tff(fact_10241_ATP_Olambda__131,axiom,
    ! [Uu2: fun_real_complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_gi(Uu2),Uua)
    <=> ( aa_real_complex(Uu2,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_131
tff(fact_10242_ATP_Olambda__132,axiom,
    ! [Uu2: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_gj(Uu2),Uua)
    <=> ( aa_real_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_132
tff(fact_10243_ATP_Olambda__133,axiom,
    ! [Uu2: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_gl(Uu2),Uua)
    <=> ( aa_real_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_133
tff(fact_10244_ATP_Olambda__134,axiom,
    ! [Uu2: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_gn(Uu2),Uua)
    <=> ( aa_real_nat(Uu2,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_134
tff(fact_10245_ATP_Olambda__135,axiom,
    ! [Uu2: fun_real_int,Uua: real] :
      ( aa_real_o(aTP_Lamm_gp(Uu2),Uua)
    <=> ( aa_real_int(Uu2,Uua) = zero_zero_int ) ) ).

% ATP.lambda_135
tff(fact_10246_ATP_Olambda__136,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_gr(Uu2),Uua)
    <=> ( aa_nat_complex(Uu2,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_136
tff(fact_10247_ATP_Olambda__137,axiom,
    ! [Uu2: fun_complex_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_rl(Uu2),Uua)
    <=> ( aa_complex_complex(Uu2,Uua) = one_one_complex ) ) ).

% ATP.lambda_137
tff(fact_10248_ATP_Olambda__138,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_rn(Uu2),Uua)
    <=> ( aa_complex_real(Uu2,Uua) = one_one_real ) ) ).

% ATP.lambda_138
tff(fact_10249_ATP_Olambda__139,axiom,
    ! [Uu2: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_rp(Uu2),Uua)
    <=> ( aa_complex_rat(Uu2,Uua) = one_one_rat ) ) ).

% ATP.lambda_139
tff(fact_10250_ATP_Olambda__140,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_rr(Uu2),Uua)
    <=> ( aa_complex_nat(Uu2,Uua) = one_one_nat ) ) ).

% ATP.lambda_140
tff(fact_10251_ATP_Olambda__141,axiom,
    ! [Uu2: fun_complex_int,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_rt(Uu2),Uua)
    <=> ( aa_complex_int(Uu2,Uua) = one_one_int ) ) ).

% ATP.lambda_141
tff(fact_10252_ATP_Olambda__142,axiom,
    ! [Uu2: fun_real_complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_rk(Uu2),Uua)
    <=> ( aa_real_complex(Uu2,Uua) = one_one_complex ) ) ).

% ATP.lambda_142
tff(fact_10253_ATP_Olambda__143,axiom,
    ! [Uu2: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_rm(Uu2),Uua)
    <=> ( aa_real_real(Uu2,Uua) = one_one_real ) ) ).

% ATP.lambda_143
tff(fact_10254_ATP_Olambda__144,axiom,
    ! [Uu2: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_ro(Uu2),Uua)
    <=> ( aa_real_rat(Uu2,Uua) = one_one_rat ) ) ).

% ATP.lambda_144
tff(fact_10255_ATP_Olambda__145,axiom,
    ! [Uu2: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_rq(Uu2),Uua)
    <=> ( aa_real_nat(Uu2,Uua) = one_one_nat ) ) ).

% ATP.lambda_145
tff(fact_10256_ATP_Olambda__146,axiom,
    ! [Uu2: fun_real_int,Uua: real] :
      ( aa_real_o(aTP_Lamm_rs(Uu2),Uua)
    <=> ( aa_real_int(Uu2,Uua) = one_one_int ) ) ).

% ATP.lambda_146
tff(fact_10257_ATP_Olambda__147,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adt(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_add(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_147
tff(fact_10258_ATP_Olambda__148,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ads(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_add(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_148
tff(fact_10259_ATP_Olambda__149,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abk,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uu2) ).

% ATP.lambda_149
tff(fact_10260_ATP_Olambda__150,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_yq(Uu2),Uua) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_150
tff(fact_10261_ATP_Olambda__151,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yp,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_151
tff(fact_10262_ATP_Olambda__152,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_yc(Uu2),Uua) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_152
tff(fact_10263_ATP_Olambda__153,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yb,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_153
tff(fact_10264_ATP_Olambda__154,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_xw(Uu2),Uua) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua)) ).

% ATP.lambda_154
tff(fact_10265_ATP_Olambda__155,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xv,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_155
tff(fact_10266_ATP_Olambda__156,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_yf(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri5044797733671781792omplex,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua)) ).

% ATP.lambda_156
tff(fact_10267_ATP_Olambda__157,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yg,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_157
tff(fact_10268_ATP_Olambda__158,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mr(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua)) ).

% ATP.lambda_158
tff(fact_10269_ATP_Olambda__159,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mq,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_159
tff(fact_10270_ATP_Olambda__160,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aTP_Lamm_abm(Uu2),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu2))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_160
tff(fact_10271_ATP_Olambda__161,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_yu(Uu2),Uua) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_nat_real(cos_coeff,Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uu2)),Uua)) ).

% ATP.lambda_161
tff(fact_10272_ATP_Olambda__162,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yt,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uu2)),Uua)) ).

% ATP.lambda_162
tff(fact_10273_ATP_Olambda__163,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_acj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu2)) ).

% ATP.lambda_163
tff(fact_10274_ATP_Olambda__164,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_yi(Uu2),Uua) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_nat_real(sin_coeff,Uua)),aa_nat_complex(power_power_complex(Uu2),Uua)) ).

% ATP.lambda_164
tff(fact_10275_ATP_Olambda__165,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yh,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_165
tff(fact_10276_ATP_Olambda__166,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_yk(Uu2),Uua) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_nat_real(cos_coeff,Uua)),aa_nat_complex(power_power_complex(Uu2),Uua)) ).

% ATP.lambda_166
tff(fact_10277_ATP_Olambda__167,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_167
tff(fact_10278_ATP_Olambda__168,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nu,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_168
tff(fact_10279_ATP_Olambda__169,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nl,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_169
tff(fact_10280_ATP_Olambda__170,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_aan,Uu2),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu2),
        $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_170
tff(fact_10281_ATP_Olambda__171,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_aao,Uu2),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu2),
        $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_171
tff(fact_10282_ATP_Olambda__172,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_oe(Uu2),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu2) ).

% ATP.lambda_172
tff(fact_10283_ATP_Olambda__173,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_og,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu2) ).

% ATP.lambda_173
tff(fact_10284_ATP_Olambda__174,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_of(Uu2),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu2) ).

% ATP.lambda_174
tff(fact_10285_ATP_Olambda__175,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_jm,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_175
tff(fact_10286_ATP_Olambda__176,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jn,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_176
tff(fact_10287_ATP_Olambda__177,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_jj,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_177
tff(fact_10288_ATP_Olambda__178,axiom,
    ! [Uu2: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jk,Uu2),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_178
tff(fact_10289_ATP_Olambda__179,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ok(Uu2),Uua) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_179
tff(fact_10290_ATP_Olambda__180,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ol,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% ATP.lambda_180
tff(fact_10291_ATP_Olambda__181,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_om(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua),numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% ATP.lambda_181
tff(fact_10292_ATP_Olambda__182,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xh,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)) ).

% ATP.lambda_182
tff(fact_10293_ATP_Olambda__183,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ado,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_183
tff(fact_10294_ATP_Olambda__184,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adm,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_184
tff(fact_10295_ATP_Olambda__185,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_185
tff(fact_10296_ATP_Olambda__186,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_um(Uu2),Uua) = aa_complex_complex(minus_minus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_186
tff(fact_10297_ATP_Olambda__187,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uo,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_187
tff(fact_10298_ATP_Olambda__188,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_un(Uu2),Uua) = minus_minus_rat(Uu2,aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_188
tff(fact_10299_ATP_Olambda__189,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_up,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_189
tff(fact_10300_ATP_Olambda__190,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_uq(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_190
tff(fact_10301_ATP_Olambda__191,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tf,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_191
tff(fact_10302_ATP_Olambda__192,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_tg(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_192
tff(fact_10303_ATP_Olambda__193,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_th,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_193
tff(fact_10304_ATP_Olambda__194,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_ti(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_194
tff(fact_10305_ATP_Olambda__195,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aeh,Uu2),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu2) ) ) ).

% ATP.lambda_195
tff(fact_10306_ATP_Olambda__196,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_qc,Uu2),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),Uu2) ) ).

% ATP.lambda_196
tff(fact_10307_ATP_Olambda__197,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_qd(Uu2),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu2) ) ).

% ATP.lambda_197
tff(fact_10308_ATP_Olambda__198,axiom,
    ! [Uu2: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_qe,Uu2),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu2) ) ).

% ATP.lambda_198
tff(fact_10309_ATP_Olambda__199,axiom,
    ! [Uu2: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_qf,Uu2),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu2) ) ).

% ATP.lambda_199
tff(fact_10310_ATP_Olambda__200,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bt,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu2) ) ).

% ATP.lambda_200
tff(fact_10311_ATP_Olambda__201,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qg,Uu2),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu2) ) ).

% ATP.lambda_201
tff(fact_10312_ATP_Olambda__202,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aci,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu2) ).

% ATP.lambda_202
tff(fact_10313_ATP_Olambda__203,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kn,Uu2),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu2) ) ).

% ATP.lambda_203
tff(fact_10314_ATP_Olambda__204,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_kj(Uu2),Uua)
    <=> aa_set_nat_o(ord_less_set_nat(Uua),Uu2) ) ).

% ATP.lambda_204
tff(fact_10315_ATP_Olambda__205,axiom,
    ! [Uu2: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_kk,Uu2),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu2) ) ).

% ATP.lambda_205
tff(fact_10316_ATP_Olambda__206,axiom,
    ! [Uu2: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_kl,Uu2),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu2) ) ).

% ATP.lambda_206
tff(fact_10317_ATP_Olambda__207,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bu,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2) ) ).

% ATP.lambda_207
tff(fact_10318_ATP_Olambda__208,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_km,Uu2),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu2) ) ).

% ATP.lambda_208
tff(fact_10319_ATP_Olambda__209,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ag,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu2) ).

% ATP.lambda_209
tff(fact_10320_ATP_Olambda__210,axiom,
    ! [Uu2: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_ah(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Uua),Uu2) ).

% ATP.lambda_210
tff(fact_10321_ATP_Olambda__211,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ai,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu2) ).

% ATP.lambda_211
tff(fact_10322_ATP_Olambda__212,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_aj(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu2) ).

% ATP.lambda_212
tff(fact_10323_ATP_Olambda__213,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aby,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu2) ).

% ATP.lambda_213
tff(fact_10324_ATP_Olambda__214,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ack,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) ).

% ATP.lambda_214
tff(fact_10325_ATP_Olambda__215,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aec,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu2) ).

% ATP.lambda_215
tff(fact_10326_ATP_Olambda__216,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_abz(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu2) ).

% ATP.lambda_216
tff(fact_10327_ATP_Olambda__217,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_acm,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu2) ).

% ATP.lambda_217
tff(fact_10328_ATP_Olambda__218,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ff,Uu2),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu2) ) ).

% ATP.lambda_218
tff(fact_10329_ATP_Olambda__219,axiom,
    ! [Uu2: complex,Uua: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_fc,Uu2),Uua)
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Uua),Uu2) ) ).

% ATP.lambda_219
tff(fact_10330_ATP_Olambda__220,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_fd,Uu2),Uua)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Uua),Uu2) ) ).

% ATP.lambda_220
tff(fact_10331_ATP_Olambda__221,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fg,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu2) ) ).

% ATP.lambda_221
tff(fact_10332_ATP_Olambda__222,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fe,Uu2),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu2) ) ).

% ATP.lambda_222
tff(fact_10333_ATP_Olambda__223,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_abb,Uu2),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu2) ).

% ATP.lambda_223
tff(fact_10334_ATP_Olambda__224,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu2) ).

% ATP.lambda_224
tff(fact_10335_ATP_Olambda__225,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adh,Uu2),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu2) ).

% ATP.lambda_225
tff(fact_10336_ATP_Olambda__226,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_aa(Uu2),Uua)
    <=> member8440522571783428010at_nat(Uua,Uu2) ) ).

% ATP.lambda_226
tff(fact_10337_ATP_Olambda__227,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ab(Uu2),Uua)
    <=> member_complex(Uua,Uu2) ) ).

% ATP.lambda_227
tff(fact_10338_ATP_Olambda__228,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ac(Uu2),Uua)
    <=> member_real(Uua,Uu2) ) ).

% ATP.lambda_228
tff(fact_10339_ATP_Olambda__229,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ad(Uu2),Uua)
    <=> member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_229
tff(fact_10340_ATP_Olambda__230,axiom,
    ! [Uu2: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ae(Uu2),Uua)
    <=> member_set_nat(Uua,Uu2) ) ).

% ATP.lambda_230
tff(fact_10341_ATP_Olambda__231,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_af(Uu2),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu2) ) ).

% ATP.lambda_231
tff(fact_10342_ATP_Olambda__232,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_a(Uu2),Uua)
    <=> member_int(Uua,Uu2) ) ).

% ATP.lambda_232
tff(fact_10343_ATP_Olambda__233,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lw(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_233
tff(fact_10344_ATP_Olambda__234,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lv(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_234
tff(fact_10345_ATP_Olambda__235,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mu(Uu2),Uua) = aa_nat_complex(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_235
tff(fact_10346_ATP_Olambda__236,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hz(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_236
tff(fact_10347_ATP_Olambda__237,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_adu(Uu2),Uua)
    <=> aa_nat_o(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_237
tff(fact_10348_ATP_Olambda__238,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ii(Uu2),Uua) = aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_238
tff(fact_10349_ATP_Olambda__239,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hy(Uu2),Uua) = aa_nat_nat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_239
tff(fact_10350_ATP_Olambda__240,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ij(Uu2),Uua) = aa_nat_int(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_240
tff(fact_10351_ATP_Olambda__241,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_abt(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_241
tff(fact_10352_ATP_Olambda__242,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_abq(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_242
tff(fact_10353_ATP_Olambda__243,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abj,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abi(Uu2),Uua)) ).

% ATP.lambda_243
tff(fact_10354_ATP_Olambda__244,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abh,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abg(Uu2),Uua)) ).

% ATP.lambda_244
tff(fact_10355_ATP_Olambda__245,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_abf,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abe(Uu2),Uua)) ).

% ATP.lambda_245
tff(fact_10356_ATP_Olambda__246,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_abd,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abc(Uu2),Uua)) ).

% ATP.lambda_246
tff(fact_10357_ATP_Olambda__247,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_aba,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aaz(Uu2),Uua)) ).

% ATP.lambda_247
tff(fact_10358_ATP_Olambda__248,axiom,
    ! [Uu2: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_act(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_acs(Uu2),Uua)) ).

% ATP.lambda_248
tff(fact_10359_ATP_Olambda__249,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_zg(Uu2),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu2))) ).

% ATP.lambda_249
tff(fact_10360_ATP_Olambda__250,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_real(aTP_Lamm_xy(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(power_power_complex(Uu2),Uua))) ).

% ATP.lambda_250
tff(fact_10361_ATP_Olambda__251,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xx,Uu2),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua))) ).

% ATP.lambda_251
tff(fact_10362_ATP_Olambda__252,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gs,Uu2),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_252
tff(fact_10363_ATP_Olambda__253,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ys(Uu2),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_nat_real(sin_coeff,Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uu2)),Uua))) ).

% ATP.lambda_253
tff(fact_10364_ATP_Olambda__254,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yr,Uu2),Uua) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uu2)),Uua))) ).

% ATP.lambda_254
tff(fact_10365_ATP_Olambda__255,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_real(aTP_Lamm_ym(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_nat_real(sin_coeff,Uua)),aa_nat_complex(power_power_complex(Uu2),Uua))) ).

% ATP.lambda_255
tff(fact_10366_ATP_Olambda__256,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yl,Uu2),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua))) ).

% ATP.lambda_256
tff(fact_10367_ATP_Olambda__257,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_real(aTP_Lamm_yo(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_nat_real(cos_coeff,Uua)),aa_nat_complex(power_power_complex(Uu2),Uua))) ).

% ATP.lambda_257
tff(fact_10368_ATP_Olambda__258,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yn,Uu2),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua))) ).

% ATP.lambda_258
tff(fact_10369_ATP_Olambda__259,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_oh(Uu2),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_259
tff(fact_10370_ATP_Olambda__260,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_oj(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_260
tff(fact_10371_ATP_Olambda__261,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_oi(Uu2),Uua) = aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(Uu2,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_261
tff(fact_10372_ATP_Olambda__262,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_on(Uu2),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_262
tff(fact_10373_ATP_Olambda__263,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_op(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_263
tff(fact_10374_ATP_Olambda__264,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_oq(Uu2),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_264
tff(fact_10375_ATP_Olambda__265,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_or(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_265
tff(fact_10376_ATP_Olambda__266,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_oo(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_266
tff(fact_10377_ATP_Olambda__267,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aej(Uu2),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)) ).

% ATP.lambda_267
tff(fact_10378_ATP_Olambda__268,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adl,Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_268
tff(fact_10379_ATP_Olambda__269,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ace,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu2),Uua)) ).

% ATP.lambda_269
tff(fact_10380_ATP_Olambda__270,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acf,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu2)) ).

% ATP.lambda_270
tff(fact_10381_ATP_Olambda__271,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aau,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu2),Uua)) ).

% ATP.lambda_271
tff(fact_10382_ATP_Olambda__272,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aat,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu2)) ).

% ATP.lambda_272
tff(fact_10383_ATP_Olambda__273,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_mg(Uu2),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu2,Uua)) ).

% ATP.lambda_273
tff(fact_10384_ATP_Olambda__274,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ts(Uu2),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_274
tff(fact_10385_ATP_Olambda__275,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adf(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_275
tff(fact_10386_ATP_Olambda__276,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_qh(Uu2),Uua) = times_times_complex(aa_nat_complex(Uu2,Uua)) ).

% ATP.lambda_276
tff(fact_10387_ATP_Olambda__277,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_qi(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_277
tff(fact_10388_ATP_Olambda__278,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_qj(Uu2),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_278
tff(fact_10389_ATP_Olambda__279,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_qk(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_279
tff(fact_10390_ATP_Olambda__280,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ql(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_280
tff(fact_10391_ATP_Olambda__281,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_tv(Uu2),Uua) = plus_plus_complex(aa_nat_complex(Uu2,Uua)) ).

% ATP.lambda_281
tff(fact_10392_ATP_Olambda__282,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_tz(Uu2),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_282
tff(fact_10393_ATP_Olambda__283,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_tw(Uu2),Uua) = aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_283
tff(fact_10394_ATP_Olambda__284,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ty(Uu2),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_284
tff(fact_10395_ATP_Olambda__285,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_tx(Uu2),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_285
tff(fact_10396_ATP_Olambda__286,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_aea(Uu2),Uua) = aa_real_real(artanh_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_286
tff(fact_10397_ATP_Olambda__287,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_adz(Uu2),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_287
tff(fact_10398_ATP_Olambda__288,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gx(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_288
tff(fact_10399_ATP_Olambda__289,axiom,
    ! [Uu2: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_gw(Uu2),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu2,Uua)) ).

% ATP.lambda_289
tff(fact_10400_ATP_Olambda__290,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adr(Uu2),Uua) = aa_real_real(cos_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_290
tff(fact_10401_ATP_Olambda__291,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_acq(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(Uu2,Uua)) ).

% ATP.lambda_291
tff(fact_10402_ATP_Olambda__292,axiom,
    ! [Uu2: list_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_at(Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = nth_VEBT_VEBT(Uu2,I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Uu2)) ) ) ).

% ATP.lambda_292
tff(fact_10403_ATP_Olambda__293,axiom,
    ! [Uu2: list_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ap(Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_complex(nth_complex(Uu2),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Uu2)) ) ) ).

% ATP.lambda_293
tff(fact_10404_ATP_Olambda__294,axiom,
    ! [Uu2: list_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aq(Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_real(nth_real(Uu2),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Uu2)) ) ) ).

% ATP.lambda_294
tff(fact_10405_ATP_Olambda__295,axiom,
    ! [Uu2: list_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ar(Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = nth_list_nat(Uu2,I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3023201423986296836st_nat(Uu2)) ) ) ).

% ATP.lambda_295
tff(fact_10406_ATP_Olambda__296,axiom,
    ! [Uu2: list_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_au(Uu2),(Uua))
    <=> ? [I3: nat] :
          ( ( (Uua)
          <=> aa_nat_o(nth_o(Uu2),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Uu2)) ) ) ).

% ATP.lambda_296
tff(fact_10407_ATP_Olambda__297,axiom,
    ! [Uu2: list_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_as(Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_set_nat(nth_set_nat(Uu2),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Uu2)) ) ) ).

% ATP.lambda_297
tff(fact_10408_ATP_Olambda__298,axiom,
    ! [Uu2: list_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_av(Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_nat(nth_nat(Uu2),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Uu2)) ) ) ).

% ATP.lambda_298
tff(fact_10409_ATP_Olambda__299,axiom,
    ! [Uu2: list_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aw(Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa_nat_int(nth_int(Uu2),I3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Uu2)) ) ) ).

% ATP.lambda_299
tff(fact_10410_ATP_Olambda__300,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_aei(Uu2),Uua) = aa_nat_set_nat(set_ord_atMost_nat,Uu2) ).

% ATP.lambda_300
tff(fact_10411_ATP_Olambda__301,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ne(Uu2),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(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_301
tff(fact_10412_ATP_Olambda__302,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lu(Uu2),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(Uu2,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_302
tff(fact_10413_ATP_Olambda__303,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_gv(Uu2),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu2)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_303
tff(fact_10414_ATP_Olambda__304,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_gt(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu2)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu2))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_304
tff(fact_10415_ATP_Olambda__305,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gu(Uu2),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu2)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu2))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_305
tff(fact_10416_ATP_Olambda__306,axiom,
    ! [Uu2: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_pi(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_306
tff(fact_10417_ATP_Olambda__307,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_pm(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_307
tff(fact_10418_ATP_Olambda__308,axiom,
    ! [Uu2: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_pg(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_308
tff(fact_10419_ATP_Olambda__309,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pk(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_309
tff(fact_10420_ATP_Olambda__310,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_po(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_310
tff(fact_10421_ATP_Olambda__311,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ph(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_311
tff(fact_10422_ATP_Olambda__312,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pl(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_312
tff(fact_10423_ATP_Olambda__313,axiom,
    ! [Uu2: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_pf(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_313
tff(fact_10424_ATP_Olambda__314,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_pj(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_314
tff(fact_10425_ATP_Olambda__315,axiom,
    ! [Uu2: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pn(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_int_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_315
tff(fact_10426_ATP_Olambda__316,axiom,
    ! [Uu2: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_oy(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_316
tff(fact_10427_ATP_Olambda__317,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_pc(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_317
tff(fact_10428_ATP_Olambda__318,axiom,
    ! [Uu2: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_ow(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_318
tff(fact_10429_ATP_Olambda__319,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pa(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_319
tff(fact_10430_ATP_Olambda__320,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_pe(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_320
tff(fact_10431_ATP_Olambda__321,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ox(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_321
tff(fact_10432_ATP_Olambda__322,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pb(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_322
tff(fact_10433_ATP_Olambda__323,axiom,
    ! [Uu2: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ov(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_323
tff(fact_10434_ATP_Olambda__324,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_oz(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_324
tff(fact_10435_ATP_Olambda__325,axiom,
    ! [Uu2: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_pd(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_int_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_325
tff(fact_10436_ATP_Olambda__326,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aax(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu2),Uub))) ).

% ATP.lambda_326
tff(fact_10437_ATP_Olambda__327,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aay(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu2),Uub))) ).

% ATP.lambda_327
tff(fact_10438_ATP_Olambda__328,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aaw(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu2,Uub))) ).

% ATP.lambda_328
tff(fact_10439_ATP_Olambda__329,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_o,Uub: complex] :
      aa_complex_complex(aa_fun3681620399087435360omplex(aTP_Lamm_qt(Uu2),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_complex(Uu2,Uub),one_one_complex) ).

% ATP.lambda_329
tff(fact_10440_ATP_Olambda__330,axiom,
    ! [Uu2: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_qx(Uu2),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu2,Uub),one_one_real) ).

% ATP.lambda_330
tff(fact_10441_ATP_Olambda__331,axiom,
    ! [Uu2: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_qp(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu2,Uub),one_one_complex) ).

% ATP.lambda_331
tff(fact_10442_ATP_Olambda__332,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_qv(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu2,Uub),one_one_real) ).

% ATP.lambda_332
tff(fact_10443_ATP_Olambda__333,axiom,
    ! [Uu2: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_qz(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu2,Uub),one_one_rat) ).

% ATP.lambda_333
tff(fact_10444_ATP_Olambda__334,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_qr(Uu2),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu2,Uub),one_one_complex) ).

% ATP.lambda_334
tff(fact_10445_ATP_Olambda__335,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_o,Uub: nat] :
      aa_nat_real(aa_fun5510454223943551838t_real(aTP_Lamm_qw(Uu2),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_real(Uu2,Uub),one_one_real) ).

% ATP.lambda_335
tff(fact_10446_ATP_Olambda__336,axiom,
    ! [Uu2: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_qn(Uu2),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu2,Uub),one_one_complex) ).

% ATP.lambda_336
tff(fact_10447_ATP_Olambda__337,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_qu(Uu2),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu2,Uub),one_one_real) ).

% ATP.lambda_337
tff(fact_10448_ATP_Olambda__338,axiom,
    ! [Uu2: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_qy(Uu2),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu2,Uub),one_one_rat) ).

% ATP.lambda_338
tff(fact_10449_ATP_Olambda__339,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tb(Uu2),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu2,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_339
tff(fact_10450_ATP_Olambda__340,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sz(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu2,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_340
tff(fact_10451_ATP_Olambda__341,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_wr(Uu2),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu2,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_341
tff(fact_10452_ATP_Olambda__342,axiom,
    ! [Uu2: fun_co1148497455491383662plex_o,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_cm(Uu2),Uua),Uub)
    <=> aa_complex_o(aa_com5808801942499186519plex_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_342
tff(fact_10453_ATP_Olambda__343,axiom,
    ! [Uu2: fun_co4998825143550818160real_o,Uua: real,Uub: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aTP_Lamm_ca(Uu2),Uua),Uub)
    <=> aa_real_o(aa_com1951324360330529881real_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_343
tff(fact_10454_ATP_Olambda__344,axiom,
    ! [Uu2: fun_co5676208753217392844_nat_o,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_cg(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_complex_fun_nat_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_344
tff(fact_10455_ATP_Olambda__345,axiom,
    ! [Uu2: fun_re7319081385389557872plex_o,Uua: complex,Uub: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aTP_Lamm_ci(Uu2),Uua),Uub)
    <=> aa_complex_o(aa_rea2382570592068293209plex_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_345
tff(fact_10456_ATP_Olambda__346,axiom,
    ! [Uu2: fun_real_fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bw(Uu2),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_346
tff(fact_10457_ATP_Olambda__347,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_acp(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_347
tff(fact_10458_ATP_Olambda__348,axiom,
    ! [Uu2: fun_real_fun_nat_o,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_cc(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_real_fun_nat_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_348
tff(fact_10459_ATP_Olambda__349,axiom,
    ! [Uu2: fun_li7444029373303974462real_o,Uua: real,Uub: list_nat] :
      ( aa_list_nat_o(aa_rea6225427375947155751_nat_o(aTP_Lamm_co(Uu2),Uua),Uub)
    <=> aa_real_o(aa_lis7606507341688705831real_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_349
tff(fact_10460_ATP_Olambda__350,axiom,
    ! [Uu2: fun_na1554462459066888140plex_o,Uua: complex,Uub: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aTP_Lamm_ck(Uu2),Uua),Uub)
    <=> aa_complex_o(aa_nat_fun_complex_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_350
tff(fact_10461_ATP_Olambda__351,axiom,
    ! [Uu2: fun_nat_fun_real_o,Uua: real,Uub: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aTP_Lamm_by(Uu2),Uua),Uub)
    <=> aa_real_o(aa_nat_fun_real_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_351
tff(fact_10462_ATP_Olambda__352,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sj(Uu2),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_352
tff(fact_10463_ATP_Olambda__353,axiom,
    ! [Uu2: fun_nat_fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ce(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(Uu2,Uub),Uua) ) ).

% ATP.lambda_353
tff(fact_10464_ATP_Olambda__354,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu2,Uub),Uua) ).

% ATP.lambda_354
tff(fact_10465_ATP_Olambda__355,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qa(Uu2),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu2,Uub),Uua) ).

% ATP.lambda_355
tff(fact_10466_ATP_Olambda__356,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_yy(Uu2),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_yx(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_356
tff(fact_10467_ATP_Olambda__357,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yw(Uu2),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yv(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_357
tff(fact_10468_ATP_Olambda__358,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_xu(Uu2),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xt(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_358
tff(fact_10469_ATP_Olambda__359,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xs(Uu2),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xr(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_359
tff(fact_10470_ATP_Olambda__360,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_xq(Uu2),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xp(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_360
tff(fact_10471_ATP_Olambda__361,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xo(Uu2),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xn(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_361
tff(fact_10472_ATP_Olambda__362,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tu(Uu2),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_tt(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_362
tff(fact_10473_ATP_Olambda__363,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tr(Uu2),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_tq(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_363
tff(fact_10474_ATP_Olambda__364,axiom,
    ! [Uu2: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mx(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex($ite(Uub = Uu2,one_one_complex,zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_364
tff(fact_10475_ATP_Olambda__365,axiom,
    ! [Uu2: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_my(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(Uub = Uu2,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_365
tff(fact_10476_ATP_Olambda__366,axiom,
    ! [Uu2: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_mz(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu2,one_one_int,zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_366
tff(fact_10477_ATP_Olambda__367,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_aav(Uu2),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu2),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_367
tff(fact_10478_ATP_Olambda__368,axiom,
    ! [Uu2: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qb(Uu2),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qa(Uu2),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_368
tff(fact_10479_ATP_Olambda__369,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_py(Uu2),Uua),Uub) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu2),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_369
tff(fact_10480_ATP_Olambda__370,axiom,
    ! [Uu2: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sk(Uu2),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sj(Uu2),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_370
tff(fact_10481_ATP_Olambda__371,axiom,
    ! [Uu2: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sh(Uu2),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu2),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_371
tff(fact_10482_ATP_Olambda__372,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aal(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aak(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_372
tff(fact_10483_ATP_Olambda__373,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aaj(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aai(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_373
tff(fact_10484_ATP_Olambda__374,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aah(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aag(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_374
tff(fact_10485_ATP_Olambda__375,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aaf(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aae(Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_375
tff(fact_10486_ATP_Olambda__376,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aad(Uu2),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aac(Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_376
tff(fact_10487_ATP_Olambda__377,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aab(Uu2),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aaa(Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_377
tff(fact_10488_ATP_Olambda__378,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acu(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_378
tff(fact_10489_ATP_Olambda__379,axiom,
    ! [Uu2: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_nn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_379
tff(fact_10490_ATP_Olambda__380,axiom,
    ! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_no(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_380
tff(fact_10491_ATP_Olambda__381,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_np(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_381
tff(fact_10492_ATP_Olambda__382,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_nq(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_382
tff(fact_10493_ATP_Olambda__383,axiom,
    ! [Uu2: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_nr(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_383
tff(fact_10494_ATP_Olambda__384,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_td(Uu2),Uua),Uub)
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_kq(Uu2),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_384
tff(fact_10495_ATP_Olambda__385,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_te(Uu2),Uua),Uub)
    <=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko(Uu2),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_385
tff(fact_10496_ATP_Olambda__386,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hb(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_386
tff(fact_10497_ATP_Olambda__387,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hd(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_387
tff(fact_10498_ATP_Olambda__388,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_hc(Uu2),Uua),Uub) = divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_388
tff(fact_10499_ATP_Olambda__389,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acr(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_389
tff(fact_10500_ATP_Olambda__390,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ns(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_390
tff(fact_10501_ATP_Olambda__391,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aef(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_391
tff(fact_10502_ATP_Olambda__392,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ng(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu2,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_392
tff(fact_10503_ATP_Olambda__393,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ni(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu2,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_393
tff(fact_10504_ATP_Olambda__394,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aeg(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_394
tff(fact_10505_ATP_Olambda__395,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bp(Uu2),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_395
tff(fact_10506_ATP_Olambda__396,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bo(Uu2),Uua),Uub)
    <=> ( ord_le211207098394363844omplex(set_complex2(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_396
tff(fact_10507_ATP_Olambda__397,axiom,
    ! [Uu2: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_bq(Uu2),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_397
tff(fact_10508_ATP_Olambda__398,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bs(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_398
tff(fact_10509_ATP_Olambda__399,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_br(Uu2),Uua),Uub)
    <=> ( ord_less_eq_set_int(set_int2(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_399
tff(fact_10510_ATP_Olambda__400,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bk(Uu2),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu2)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_400
tff(fact_10511_ATP_Olambda__401,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bj(Uu2),Uua),Uub)
    <=> ( ord_le211207098394363844omplex(set_complex2(Uub),Uu2)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_401
tff(fact_10512_ATP_Olambda__402,axiom,
    ! [Uu2: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_bl(Uu2),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu2)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_402
tff(fact_10513_ATP_Olambda__403,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bn(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu2)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_403
tff(fact_10514_ATP_Olambda__404,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bm(Uu2),Uua),Uub)
    <=> ( ord_less_eq_set_int(set_int2(Uub),Uu2)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_404
tff(fact_10515_ATP_Olambda__405,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aee(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_405
tff(fact_10516_ATP_Olambda__406,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aap(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_406
tff(fact_10517_ATP_Olambda__407,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ael(Uu2),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_407
tff(fact_10518_ATP_Olambda__408,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aaq(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_408
tff(fact_10519_ATP_Olambda__409,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_nf(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu2),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_409
tff(fact_10520_ATP_Olambda__410,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nh(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu2),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_410
tff(fact_10521_ATP_Olambda__411,axiom,
    ! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nk(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu2),Uub)) ).

% ATP.lambda_411
tff(fact_10522_ATP_Olambda__412,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nj(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_412
tff(fact_10523_ATP_Olambda__413,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wo(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uub)) ).

% ATP.lambda_413
tff(fact_10524_ATP_Olambda__414,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ach(Uu2),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),Uu2) ) ) ).

% ATP.lambda_414
tff(fact_10525_ATP_Olambda__415,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_qs(Uu2),Uua),Uub)
    <=> ( member_complex(Uub,Uu2)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_415
tff(fact_10526_ATP_Olambda__416,axiom,
    ! [Uu2: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_qo(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_416
tff(fact_10527_ATP_Olambda__417,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_qq(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_417
tff(fact_10528_ATP_Olambda__418,axiom,
    ! [Uu2: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_qm(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_418
tff(fact_10529_ATP_Olambda__419,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_eo(Uu2),Uua),Uub)
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_419
tff(fact_10530_ATP_Olambda__420,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ew(Uu2),Uua),Uub)
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_420
tff(fact_10531_ATP_Olambda__421,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_ek(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_421
tff(fact_10532_ATP_Olambda__422,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_es(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_422
tff(fact_10533_ATP_Olambda__423,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_fa(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_423
tff(fact_10534_ATP_Olambda__424,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_em(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_424
tff(fact_10535_ATP_Olambda__425,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_eu(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_425
tff(fact_10536_ATP_Olambda__426,axiom,
    ! [Uu2: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_ei(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_426
tff(fact_10537_ATP_Olambda__427,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_eq(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_427
tff(fact_10538_ATP_Olambda__428,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ey(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_428
tff(fact_10539_ATP_Olambda__429,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_du(Uu2),Uua),Uub)
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_429
tff(fact_10540_ATP_Olambda__430,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ec(Uu2),Uua),Uub)
    <=> ( member_complex(Uub,Uu2)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_430
tff(fact_10541_ATP_Olambda__431,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_dq(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_431
tff(fact_10542_ATP_Olambda__432,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_dy(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_432
tff(fact_10543_ATP_Olambda__433,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_eg(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_433
tff(fact_10544_ATP_Olambda__434,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_ds(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_434
tff(fact_10545_ATP_Olambda__435,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_ea(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_435
tff(fact_10546_ATP_Olambda__436,axiom,
    ! [Uu2: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_do(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_436
tff(fact_10547_ATP_Olambda__437,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_dw(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_437
tff(fact_10548_ATP_Olambda__438,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ee(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ( aa_int_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_438
tff(fact_10549_ATP_Olambda__439,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_ay(Uu2),Uua),Uub)
    <=> ( member8440522571783428010at_nat(Uub,Uu2)
        & ~ member8440522571783428010at_nat(Uub,Uua) ) ) ).

% ATP.lambda_439
tff(fact_10550_ATP_Olambda__440,axiom,
    ! [Uu2: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_az(Uu2),Uua),Uub)
    <=> ( member_complex(Uub,Uu2)
        & ~ member_complex(Uub,Uua) ) ) ).

% ATP.lambda_440
tff(fact_10551_ATP_Olambda__441,axiom,
    ! [Uu2: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_ba(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ~ member_real(Uub,Uua) ) ) ).

% ATP.lambda_441
tff(fact_10552_ATP_Olambda__442,axiom,
    ! [Uu2: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_bb(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_442
tff(fact_10553_ATP_Olambda__443,axiom,
    ! [Uu2: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_bc(Uu2),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu2)
        & ~ member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_443
tff(fact_10554_ATP_Olambda__444,axiom,
    ! [Uu2: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bd(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_444
tff(fact_10555_ATP_Olambda__445,axiom,
    ! [Uu2: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ax(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ~ member_int(Uub,Uua) ) ) ).

% ATP.lambda_445
tff(fact_10556_ATP_Olambda__446,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acg(Uu2),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)),Uu2) ) ).

% ATP.lambda_446
tff(fact_10557_ATP_Olambda__447,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sy(Uu2),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)),Uu2) ) ).

% ATP.lambda_447
tff(fact_10558_ATP_Olambda__448,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_tp(Uu2),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)),Uu2) ) ).

% ATP.lambda_448
tff(fact_10559_ATP_Olambda__449,axiom,
    ! [Uu2: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_ml(Uu2),Uua),(Uub))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(Uua,(Uub)),Uu2) ) ).

% ATP.lambda_449
tff(fact_10560_ATP_Olambda__450,axiom,
    ! [Uu2: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_mm(Uu2),Uua),Uub)
    <=> member7279096912039735102um_num(product_Pair_num_num(Uua,Uub),Uu2) ) ).

% ATP.lambda_450
tff(fact_10561_ATP_Olambda__451,axiom,
    ! [Uu2: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_mn(Uu2),Uua),Uub)
    <=> member9148766508732265716at_num(product_Pair_nat_num(Uua,Uub),Uu2) ) ).

% ATP.lambda_451
tff(fact_10562_ATP_Olambda__452,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_mo(Uu2),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu2) ) ).

% ATP.lambda_452
tff(fact_10563_ATP_Olambda__453,axiom,
    ! [Uu2: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_mp(Uu2),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu2) ) ).

% ATP.lambda_453
tff(fact_10564_ATP_Olambda__454,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_dn(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu2) = Uua ) ) ).

% ATP.lambda_454
tff(fact_10565_ATP_Olambda__455,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_zy(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_455
tff(fact_10566_ATP_Olambda__456,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_mc(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_456
tff(fact_10567_ATP_Olambda__457,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_md(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_457
tff(fact_10568_ATP_Olambda__458,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_adq(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_458
tff(fact_10569_ATP_Olambda__459,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acs(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_459
tff(fact_10570_ATP_Olambda__460,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_kq(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_460
tff(fact_10571_ATP_Olambda__461,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ko(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_461
tff(fact_10572_ATP_Olambda__462,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_tj(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_462
tff(fact_10573_ATP_Olambda__463,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uj(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_463
tff(fact_10574_ATP_Olambda__464,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_tk(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_464
tff(fact_10575_ATP_Olambda__465,axiom,
    ! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dl(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_465
tff(fact_10576_ATP_Olambda__466,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pp(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_466
tff(fact_10577_ATP_Olambda__467,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pq(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_467
tff(fact_10578_ATP_Olambda__468,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_pr(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_468
tff(fact_10579_ATP_Olambda__469,axiom,
    ! [Uu2: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_iq(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu2,Uub)) ).

% ATP.lambda_469
tff(fact_10580_ATP_Olambda__470,axiom,
    ! [Uu2: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_it(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu2,Uub)) ).

% ATP.lambda_470
tff(fact_10581_ATP_Olambda__471,axiom,
    ! [Uu2: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_ip(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu2,Uub)) ).

% ATP.lambda_471
tff(fact_10582_ATP_Olambda__472,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_is(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu2,Uub)) ).

% ATP.lambda_472
tff(fact_10583_ATP_Olambda__473,axiom,
    ! [Uu2: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_iw(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uua,Uub)),aa_real_rat(Uu2,Uub)) ).

% ATP.lambda_473
tff(fact_10584_ATP_Olambda__474,axiom,
    ! [Uu2: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_in(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu2,Uub)) ).

% ATP.lambda_474
tff(fact_10585_ATP_Olambda__475,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_iu(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu2,Uub)) ).

% ATP.lambda_475
tff(fact_10586_ATP_Olambda__476,axiom,
    ! [Uu2: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_io(Uu2),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu2,Uub)) ).

% ATP.lambda_476
tff(fact_10587_ATP_Olambda__477,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ir(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu2,Uub)) ).

% ATP.lambda_477
tff(fact_10588_ATP_Olambda__478,axiom,
    ! [Uu2: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_iv(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uub)),aa_int_rat(Uu2,Uub)) ).

% ATP.lambda_478
tff(fact_10589_ATP_Olambda__479,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_gc(Uu2),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_479
tff(fact_10590_ATP_Olambda__480,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gd(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_480
tff(fact_10591_ATP_Olambda__481,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gb(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_481
tff(fact_10592_ATP_Olambda__482,axiom,
    ! [Uu2: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_hw(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu2,Uub)) ).

% ATP.lambda_482
tff(fact_10593_ATP_Olambda__483,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_hv(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu2,Uub)) ).

% ATP.lambda_483
tff(fact_10594_ATP_Olambda__484,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_hu(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu2,Uub)) ).

% ATP.lambda_484
tff(fact_10595_ATP_Olambda__485,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_na(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu2,Uub)) ).

% ATP.lambda_485
tff(fact_10596_ATP_Olambda__486,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hx(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu2,Uub)) ).

% ATP.lambda_486
tff(fact_10597_ATP_Olambda__487,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ht(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu2,Uub)) ).

% ATP.lambda_487
tff(fact_10598_ATP_Olambda__488,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fy(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_488
tff(fact_10599_ATP_Olambda__489,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ga(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_489
tff(fact_10600_ATP_Olambda__490,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fz(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_490
tff(fact_10601_ATP_Olambda__491,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ma(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_491
tff(fact_10602_ATP_Olambda__492,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fx(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_492
tff(fact_10603_ATP_Olambda__493,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aco(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_493
tff(fact_10604_ATP_Olambda__494,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dm(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu2,Uub)),Uua) ) ).

% ATP.lambda_494
tff(fact_10605_ATP_Olambda__495,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gh(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_495
tff(fact_10606_ATP_Olambda__496,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_pv(Uu2),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_496
tff(fact_10607_ATP_Olambda__497,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_gg(Uu2),Uua),Uub) = aa_int_int(modulo_modulo_int(aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_497
tff(fact_10608_ATP_Olambda__498,axiom,
    ! [Uu2: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ge(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_498
tff(fact_10609_ATP_Olambda__499,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_lx(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_499
tff(fact_10610_ATP_Olambda__500,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gf(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_500
tff(fact_10611_ATP_Olambda__501,axiom,
    ! [Uu2: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_fq(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_501
tff(fact_10612_ATP_Olambda__502,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fs(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_502
tff(fact_10613_ATP_Olambda__503,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fr(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_503
tff(fact_10614_ATP_Olambda__504,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_fp(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_504
tff(fact_10615_ATP_Olambda__505,axiom,
    ! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ms(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu2) ).

% ATP.lambda_505
tff(fact_10616_ATP_Olambda__506,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mt(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu2) ).

% ATP.lambda_506
tff(fact_10617_ATP_Olambda__507,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kw(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_507
tff(fact_10618_ATP_Olambda__508,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_kv(Uu2),Uua),Uub) = minus_minus_rat(aa_nat_rat(Uu2,Uub),Uua) ).

% ATP.lambda_508
tff(fact_10619_ATP_Olambda__509,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ku(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_509
tff(fact_10620_ATP_Olambda__510,axiom,
    ! [Uu2: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_acl(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_510
tff(fact_10621_ATP_Olambda__511,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ps(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_511
tff(fact_10622_ATP_Olambda__512,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pt(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_512
tff(fact_10623_ATP_Olambda__513,axiom,
    ! [Uu2: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_pu(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_513
tff(fact_10624_ATP_Olambda__514,axiom,
    ! [Uu2: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_adx(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu2) ).

% ATP.lambda_514
tff(fact_10625_ATP_Olambda__515,axiom,
    ! [Uu2: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_acn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_515
tff(fact_10626_ATP_Olambda__516,axiom,
    ! [Uu2: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_mi(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_516
tff(fact_10627_ATP_Olambda__517,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ye(Uu2),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)))),aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ).

% ATP.lambda_517
tff(fact_10628_ATP_Olambda__518,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_yd(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ).

% ATP.lambda_518
tff(fact_10629_ATP_Olambda__519,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jo(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_519
tff(fact_10630_ATP_Olambda__520,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_js(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_520
tff(fact_10631_ATP_Olambda__521,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_jq(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_521
tff(fact_10632_ATP_Olambda__522,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jr(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_522
tff(fact_10633_ATP_Olambda__523,axiom,
    ! [Uu2: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_jp(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_523
tff(fact_10634_ATP_Olambda__524,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_lq(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_524
tff(fact_10635_ATP_Olambda__525,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lt(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_525
tff(fact_10636_ATP_Olambda__526,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_lr(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_526
tff(fact_10637_ATP_Olambda__527,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ls(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_527
tff(fact_10638_ATP_Olambda__528,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adk(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_528
tff(fact_10639_ATP_Olambda__529,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ic(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_529
tff(fact_10640_ATP_Olambda__530,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_if(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_530
tff(fact_10641_ATP_Olambda__531,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_id(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_531
tff(fact_10642_ATP_Olambda__532,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ie(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_532
tff(fact_10643_ATP_Olambda__533,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_533
tff(fact_10644_ATP_Olambda__534,axiom,
    ! [Uu2: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fu(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(Uu2),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_534
tff(fact_10645_ATP_Olambda__535,axiom,
    ! [Uu2: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ly(Uu2),Uua),Uub) = aa_complex_complex(times_times_complex(Uu2),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_535
tff(fact_10646_ATP_Olambda__536,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fw(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_536
tff(fact_10647_ATP_Olambda__537,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fv(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_537
tff(fact_10648_ATP_Olambda__538,axiom,
    ! [Uu2: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ft(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),aa_int_int(Uua,Uub)) ).

% ATP.lambda_538
tff(fact_10649_ATP_Olambda__539,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lz(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu2,Uub)) ).

% ATP.lambda_539
tff(fact_10650_ATP_Olambda__540,axiom,
    ! [Uu2: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_rb(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_540
tff(fact_10651_ATP_Olambda__541,axiom,
    ! [Uu2: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_ra(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_541
tff(fact_10652_ATP_Olambda__542,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rc(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_542
tff(fact_10653_ATP_Olambda__543,axiom,
    ! [Uu2: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_rd(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_543
tff(fact_10654_ATP_Olambda__544,axiom,
    ! [Uu2: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_re(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_544
tff(fact_10655_ATP_Olambda__545,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tl(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_545
tff(fact_10656_ATP_Olambda__546,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_tm(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_546
tff(fact_10657_ATP_Olambda__547,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tn(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_547
tff(fact_10658_ATP_Olambda__548,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_to(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_548
tff(fact_10659_ATP_Olambda__549,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_kt(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_549
tff(fact_10660_ATP_Olambda__550,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_550
tff(fact_10661_ATP_Olambda__551,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ru(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_551
tff(fact_10662_ATP_Olambda__552,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_mh(Uu2),Uua),Uub) = aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_552
tff(fact_10663_ATP_Olambda__553,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ib(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_553
tff(fact_10664_ATP_Olambda__554,axiom,
    ! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adv(Uu2),Uua),Uub)
    <=> aa_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_554
tff(fact_10665_ATP_Olambda__555,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ia(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_555
tff(fact_10666_ATP_Olambda__556,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_rf(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_556
tff(fact_10667_ATP_Olambda__557,axiom,
    ! [Uu2: fun_Pr2602117396471556196at_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ho(Uu2),Uua),Uub) = aa_Pro4243173770044397973at_nat(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_557
tff(fact_10668_ATP_Olambda__558,axiom,
    ! [Uu2: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_hp(Uu2),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_558
tff(fact_10669_ATP_Olambda__559,axiom,
    ! [Uu2: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_hq(Uu2),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_559
tff(fact_10670_ATP_Olambda__560,axiom,
    ! [Uu2: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_hr(Uu2),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_560
tff(fact_10671_ATP_Olambda__561,axiom,
    ! [Uu2: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hs(Uu2),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_561
tff(fact_10672_ATP_Olambda__562,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mv(Uu2),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu2)) ).

% ATP.lambda_562
tff(fact_10673_ATP_Olambda__563,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mw(Uu2),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu2)) ).

% ATP.lambda_563
tff(fact_10674_ATP_Olambda__564,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_nat_real,Uub: complex] : aa_complex_real(aa_fun2896631202927662350x_real(aTP_Lamm_zr(Uu2),Uua),Uub) = aa_nat_real(Uua,aa_complex_nat(Uu2,Uub)) ).

% ATP.lambda_564
tff(fact_10675_ATP_Olambda__565,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_nat_nat,Uub: complex] : aa_complex_nat(aa_fun8118263755821012566ex_nat(aTP_Lamm_zo(Uu2),Uua),Uub) = aa_nat_nat(Uua,aa_complex_nat(Uu2,Uub)) ).

% ATP.lambda_565
tff(fact_10676_ATP_Olambda__566,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_nat_real,Uub: real] : aa_real_real(aa_fun6767034839249511948l_real(aTP_Lamm_zq(Uu2),Uua),Uub) = aa_nat_real(Uua,aa_real_nat(Uu2,Uub)) ).

% ATP.lambda_566
tff(fact_10677_ATP_Olambda__567,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_nat_nat,Uub: real] : aa_real_nat(aa_fun6786953957667191892al_nat(aTP_Lamm_zn(Uu2),Uua),Uub) = aa_nat_nat(Uua,aa_real_nat(Uu2,Uub)) ).

% ATP.lambda_567
tff(fact_10678_ATP_Olambda__568,axiom,
    ! [Uu2: fun_int_complex,Uua: fun_complex_int,Uub: int] : aa_int_int(aa_fun4384185347138573674nt_int(aTP_Lamm_zt(Uu2),Uua),Uub) = aa_complex_int(Uua,aa_int_complex(Uu2,Uub)) ).

% ATP.lambda_568
tff(fact_10679_ATP_Olambda__569,axiom,
    ! [Uu2: fun_int_real,Uua: fun_real_int,Uub: int] : aa_int_int(aa_fun4720285486439830120nt_int(aTP_Lamm_zs(Uu2),Uua),Uub) = aa_real_int(Uua,aa_int_real(Uu2,Uub)) ).

% ATP.lambda_569
tff(fact_10680_ATP_Olambda__570,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_nat_real,Uub: int] : aa_int_real(aa_fun4026127921582108556t_real(aTP_Lamm_zp(Uu2),Uua),Uub) = aa_nat_real(Uua,aa_int_nat(Uu2,Uub)) ).

% ATP.lambda_570
tff(fact_10681_ATP_Olambda__571,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_nat_nat,Uub: int] : aa_int_nat(aa_fun6790803225114646484nt_nat(aTP_Lamm_zm(Uu2),Uua),Uub) = aa_nat_nat(Uua,aa_int_nat(Uu2,Uub)) ).

% ATP.lambda_571
tff(fact_10682_ATP_Olambda__572,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_nat_int,Uub: int] : aa_int_int(aa_fun6564396834021159052nt_int(aTP_Lamm_zu(Uu2),Uua),Uub) = aa_nat_int(Uua,aa_int_nat(Uu2,Uub)) ).

% ATP.lambda_572
tff(fact_10683_ATP_Olambda__573,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_zv(Uu2),Uua),Uub) = aa_int_int(Uua,aa_int_int(Uu2,Uub)) ).

% ATP.lambda_573
tff(fact_10684_ATP_Olambda__574,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_kr(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_574
tff(fact_10685_ATP_Olambda__575,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kp(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_575
tff(fact_10686_ATP_Olambda__576,axiom,
    ! [Uu2: fun_Pr4050773816208733250omplex,Uua: fun_Pr4050773816208733250omplex,Uub: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aa_fun6819693891783611194t_real(aTP_Lamm_ss(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_Pro82899191565270131omplex(Uu2,Uub)),aa_Pro82899191565270131omplex(Uua,Uub))) ).

% ATP.lambda_576
tff(fact_10687_ATP_Olambda__577,axiom,
    ! [Uu2: fun_Pr672294278902479040t_real,Uua: fun_Pr672294278902479040t_real,Uub: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aa_fun8712866722335801276t_real(aTP_Lamm_so(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_Pro6168199777849092593t_real(Uu2,Uub)),aa_Pro6168199777849092593t_real(Uua,Uub))) ).

% ATP.lambda_577
tff(fact_10688_ATP_Olambda__578,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_real(aa_fun1953249243402654830x_real(aTP_Lamm_sr(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uub))) ).

% ATP.lambda_578
tff(fact_10689_ATP_Olambda__579,axiom,
    ! [Uu2: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_sn(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(Uu2,Uub)),aa_complex_real(Uua,Uub))) ).

% ATP.lambda_579
tff(fact_10690_ATP_Olambda__580,axiom,
    ! [Uu2: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_real(aa_fun2786211375485122666l_real(aTP_Lamm_sq(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(Uu2,Uub)),aa_real_complex(Uua,Uub))) ).

% ATP.lambda_580
tff(fact_10691_ATP_Olambda__581,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sm(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub))) ).

% ATP.lambda_581
tff(fact_10692_ATP_Olambda__582,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_su(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu2,Uub)),aa_nat_complex(Uua,Uub))) ).

% ATP.lambda_582
tff(fact_10693_ATP_Olambda__583,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_st(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub))) ).

% ATP.lambda_583
tff(fact_10694_ATP_Olambda__584,axiom,
    ! [Uu2: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_real(aa_fun1985759105522019690t_real(aTP_Lamm_sp(Uu2),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_int_complex(Uu2,Uub)),aa_int_complex(Uua,Uub))) ).

% ATP.lambda_584
tff(fact_10695_ATP_Olambda__585,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_sl(Uu2),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(Uu2,Uub)),aa_int_real(Uua,Uub))) ).

% ATP.lambda_585
tff(fact_10696_ATP_Olambda__586,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adp(Uu2),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua)) ).

% ATP.lambda_586
tff(fact_10697_ATP_Olambda__587,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_cz(Uu2),Uua),Uub)
    <=> ? [X3: complex] :
          ( ( Uub = aa_complex_complex(Uua,X3) )
          & aa_complex_o(Uu2,X3) ) ) ).

% ATP.lambda_587
tff(fact_10698_ATP_Olambda__588,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_real,Uub: real] :
      ( aa_real_o(aa_fun6876059748394487152real_o(aTP_Lamm_cx(Uu2),Uua),Uub)
    <=> ? [X3: complex] :
          ( ( Uub = aa_complex_real(Uua,X3) )
          & aa_complex_o(Uu2,X3) ) ) ).

% ATP.lambda_588
tff(fact_10699_ATP_Olambda__589,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8801047744250750888_nat_o(aTP_Lamm_cy(Uu2),Uua),Uub)
    <=> ? [X3: complex] :
          ( ( Uub = aa_complex_nat(Uua,X3) )
          & aa_complex_o(Uu2,X3) ) ) ).

% ATP.lambda_589
tff(fact_10700_ATP_Olambda__590,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_complex,Uub: complex] :
      ( aa_complex_o(aa_fun1011824800444484462plex_o(aTP_Lamm_ct(Uu2),Uua),Uub)
    <=> ? [X3: real] :
          ( ( Uub = aa_real_complex(Uua,X3) )
          & aa_real_o(Uu2,X3) ) ) ).

% ATP.lambda_590
tff(fact_10701_ATP_Olambda__591,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_cr(Uu2),Uua),Uub)
    <=> ? [X3: real] :
          ( ( Uub = aa_real_real(Uua,X3) )
          & aa_real_o(Uu2,X3) ) ) ).

% ATP.lambda_591
tff(fact_10702_ATP_Olambda__592,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun5562329742113357962_nat_o(aTP_Lamm_da(Uu2),Uua),Uub)
    <=> ? [X3: real] :
          ( ( Uub = aa_real_list_nat(Uua,X3) )
          & aa_real_o(Uu2,X3) ) ) ).

% ATP.lambda_592
tff(fact_10703_ATP_Olambda__593,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_nat,Uub: nat] :
      ( aa_nat_o(aa_fun2333486890442116906_nat_o(aTP_Lamm_cs(Uu2),Uua),Uub)
    <=> ? [X3: real] :
          ( ( Uub = aa_real_nat(Uua,X3) )
          & aa_real_o(Uu2,X3) ) ) ).

% ATP.lambda_593
tff(fact_10704_ATP_Olambda__594,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_complex,Uub: complex] :
      ( aa_complex_o(aa_fun2681562978987535434plex_o(aTP_Lamm_cw(Uu2),Uua),Uub)
    <=> ? [X3: nat] :
          ( ( Uub = aa_nat_complex(Uua,X3) )
          & aa_nat_o(Uu2,X3) ) ) ).

% ATP.lambda_594
tff(fact_10705_ATP_Olambda__595,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_real,Uub: real] :
      ( aa_real_o(aa_fun6953832176698369998real_o(aTP_Lamm_cu(Uu2),Uua),Uub)
    <=> ? [X3: nat] :
          ( ( Uub = aa_nat_real(Uua,X3) )
          & aa_nat_o(Uu2,X3) ) ) ).

% ATP.lambda_595
tff(fact_10706_ATP_Olambda__596,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_cv(Uu2),Uua),Uub)
    <=> ? [X3: nat] :
          ( ( Uub = aa_nat_nat(Uua,X3) )
          & aa_nat_o(Uu2,X3) ) ) ).

% ATP.lambda_596
tff(fact_10707_ATP_Olambda__597,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_co1148497455491383662plex_o,Uub: complex] :
      ( aa_complex_o(aa_fun7218396518982330257plex_o(aTP_Lamm_cl(Uu2),Uua),Uub)
    <=> ? [Y5: complex] :
          ( aa_complex_o(Uu2,Y5)
          & aa_complex_o(aa_com5808801942499186519plex_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_597
tff(fact_10708_ATP_Olambda__598,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_re7319081385389557872plex_o,Uub: real] :
      ( aa_real_o(aa_fun7329550764591405201real_o(aTP_Lamm_ch(Uu2),Uua),Uub)
    <=> ? [Y5: complex] :
          ( aa_complex_o(Uu2,Y5)
          & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_598
tff(fact_10709_ATP_Olambda__599,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_na1554462459066888140plex_o,Uub: nat] :
      ( aa_nat_o(aa_fun5940009423621039633_nat_o(aTP_Lamm_cj(Uu2),Uua),Uub)
    <=> ? [Y5: complex] :
          ( aa_complex_o(Uu2,Y5)
          & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_599
tff(fact_10710_ATP_Olambda__600,axiom,
    ! [Uu2: fun_real_o,Uua: fun_co4998825143550818160real_o,Uub: complex] :
      ( aa_complex_o(aa_fun4501470310227681423plex_o(aTP_Lamm_bz(Uu2),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu2,Y5)
          & aa_real_o(aa_com1951324360330529881real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_600
tff(fact_10711_ATP_Olambda__601,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun8588574755773516687real_o(aTP_Lamm_bv(Uu2),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu2,Y5)
          & aa_real_o(aa_real_fun_real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_601
tff(fact_10712_ATP_Olambda__602,axiom,
    ! [Uu2: fun_real_o,Uua: fun_li7444029373303974462real_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun5611870488384741519_nat_o(aTP_Lamm_cn(Uu2),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu2,Y5)
          & aa_real_o(aa_lis7606507341688705831real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_602
tff(fact_10713_ATP_Olambda__603,axiom,
    ! [Uu2: fun_real_o,Uua: fun_nat_fun_real_o,Uub: nat] :
      ( aa_nat_o(aa_fun1522457866930155023_nat_o(aTP_Lamm_bx(Uu2),Uua),Uub)
    <=> ? [Y5: real] :
          ( aa_real_o(Uu2,Y5)
          & aa_real_o(aa_nat_fun_real_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_603
tff(fact_10714_ATP_Olambda__604,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_co5676208753217392844_nat_o,Uub: complex] :
      ( aa_complex_o(aa_fun3010802591474140851plex_o(aTP_Lamm_cf(Uu2),Uua),Uub)
    <=> ? [Y5: nat] :
          ( aa_nat_o(Uu2,Y5)
          & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_604
tff(fact_10715_ATP_Olambda__605,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_real_fun_nat_o,Uub: real] :
      ( aa_real_o(aa_fun8982401453414893235real_o(aTP_Lamm_cb(Uu2),Uua),Uub)
    <=> ? [Y5: nat] :
          ( aa_nat_o(Uu2,Y5)
          & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_605
tff(fact_10716_ATP_Olambda__606,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun6854469565811772723_nat_o(aTP_Lamm_cd(Uu2),Uua),Uub)
    <=> ? [Y5: nat] :
          ( aa_nat_o(Uu2,Y5)
          & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Y5) ) ) ).

% ATP.lambda_606
tff(fact_10717_ATP_Olambda__607,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_yx(Uu2),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_607
tff(fact_10718_ATP_Olambda__608,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yv(Uu2),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_608
tff(fact_10719_ATP_Olambda__609,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xp(Uu2),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_609
tff(fact_10720_ATP_Olambda__610,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xn(Uu2),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_610
tff(fact_10721_ATP_Olambda__611,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xt(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_complex) ).

% ATP.lambda_611
tff(fact_10722_ATP_Olambda__612,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xr(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_real) ).

% ATP.lambda_612
tff(fact_10723_ATP_Olambda__613,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_ur(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_613
tff(fact_10724_ATP_Olambda__614,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_uv(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_614
tff(fact_10725_ATP_Olambda__615,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_us(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_615
tff(fact_10726_ATP_Olambda__616,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uu(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_616
tff(fact_10727_ATP_Olambda__617,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ut(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu2,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_617
tff(fact_10728_ATP_Olambda__618,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rw(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_618
tff(fact_10729_ATP_Olambda__619,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ry(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_619
tff(fact_10730_ATP_Olambda__620,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sa(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_620
tff(fact_10731_ATP_Olambda__621,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sc(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_621
tff(fact_10732_ATP_Olambda__622,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_se(Uu2),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu2,one_one_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_622
tff(fact_10733_ATP_Olambda__623,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_rx(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_623
tff(fact_10734_ATP_Olambda__624,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_rz(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_624
tff(fact_10735_ATP_Olambda__625,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sb(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_625
tff(fact_10736_ATP_Olambda__626,axiom,
    ! [Uu2: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sd(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_626
tff(fact_10737_ATP_Olambda__627,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sf(Uu2),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu2),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_627
tff(fact_10738_ATP_Olambda__628,axiom,
    ! [Uu2: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_nb(Uu2),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu2,Uuc)) ).

% ATP.lambda_628
tff(fact_10739_ATP_Olambda__629,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ki(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_kh(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_629
tff(fact_10740_ATP_Olambda__630,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kg(Uu2),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kf(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_630
tff(fact_10741_ATP_Olambda__631,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ke(Uu2),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kd(Uu2),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_631
tff(fact_10742_ATP_Olambda__632,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_fo(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_fn(Uu2),Uua),Uuc),Uub) ).

% ATP.lambda_632
tff(fact_10743_ATP_Olambda__633,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_fm(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_fl(Uu2),Uua),Uuc)),Uub) ).

% ATP.lambda_633
tff(fact_10744_ATP_Olambda__634,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_fk(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_fj(Uu2),Uua),Uuc),Uub) ).

% ATP.lambda_634
tff(fact_10745_ATP_Olambda__635,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_fi(Uu2),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_fh(Uu2),Uua),Uuc),Uub) ).

% ATP.lambda_635
tff(fact_10746_ATP_Olambda__636,axiom,
    ! [Uu2: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_vf(Uu2),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          $ite(
            Uuc = zero_zero_nat,
            aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),
            $ite(Uuc = Uu2,one_one_Code_integer,zero_z3403309356797280102nteger) )),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_636
tff(fact_10747_ATP_Olambda__637,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_vd(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex($ite(
            Uuc = zero_zero_nat,
            aa_complex_complex(uminus1482373934393186551omplex,Uub),
            $ite(Uuc = Uu2,one_one_complex,zero_zero_complex) )),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_637
tff(fact_10748_ATP_Olambda__638,axiom,
    ! [Uu2: nat,Uua: real,Uub: real,Uuc: nat] :
      aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_vg(Uu2),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 = Uu2,one_one_real,zero_zero_real) )),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_638
tff(fact_10749_ATP_Olambda__639,axiom,
    ! [Uu2: nat,Uua: rat,Uub: rat,Uuc: nat] :
      aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ve(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,
          $ite(
            Uuc = zero_zero_nat,
            uminus_uminus_rat(Uub),
            $ite(Uuc = Uu2,one_one_rat,zero_zero_rat) )),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_639
tff(fact_10750_ATP_Olambda__640,axiom,
    ! [Uu2: nat,Uua: int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_vc(Uu2),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 = Uu2,one_one_int,zero_zero_int) )),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_640
tff(fact_10751_ATP_Olambda__641,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ul(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_uk(Uu2),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_641
tff(fact_10752_ATP_Olambda__642,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ue(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_ud(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_642
tff(fact_10753_ATP_Olambda__643,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_uc(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ub(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_643
tff(fact_10754_ATP_Olambda__644,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_uf(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_tt(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_644
tff(fact_10755_ATP_Olambda__645,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_ua(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_tq(Uu2),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_645
tff(fact_10756_ATP_Olambda__646,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_acz(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_646
tff(fact_10757_ATP_Olambda__647,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_acx(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_647
tff(fact_10758_ATP_Olambda__648,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acv(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_648
tff(fact_10759_ATP_Olambda__649,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_acw(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_649
tff(fact_10760_ATP_Olambda__650,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_la(Uu2),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uub)) ).

% ATP.lambda_650
tff(fact_10761_ATP_Olambda__651,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_lg(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_651
tff(fact_10762_ATP_Olambda__652,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_lc(Uu2),Uua),Uub),Uuc) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_nat_rat(power_power_rat(Uu2),Uub)) ).

% ATP.lambda_652
tff(fact_10763_ATP_Olambda__653,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_le(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),Uuc))),aa_nat_int(power_power_int(Uu2),Uub)) ).

% ATP.lambda_653
tff(fact_10764_ATP_Olambda__654,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aaz(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_654
tff(fact_10765_ATP_Olambda__655,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kd(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu2)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_655
tff(fact_10766_ATP_Olambda__656,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_kh(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu2)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_656
tff(fact_10767_ATP_Olambda__657,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kf(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uua),Uu2)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_657
tff(fact_10768_ATP_Olambda__658,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xa(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_658
tff(fact_10769_ATP_Olambda__659,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_wz(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu2),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_659
tff(fact_10770_ATP_Olambda__660,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wy(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uu2),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_660
tff(fact_10771_ATP_Olambda__661,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_wu(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_661
tff(fact_10772_ATP_Olambda__662,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wx(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_662
tff(fact_10773_ATP_Olambda__663,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ww(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu2),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_663
tff(fact_10774_ATP_Olambda__664,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wt(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_664
tff(fact_10775_ATP_Olambda__665,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_wv(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_665
tff(fact_10776_ATP_Olambda__666,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ya(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_complex(power_power_complex(Uu2),Uuc))),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_666
tff(fact_10777_ATP_Olambda__667,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xz(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc))),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_667
tff(fact_10778_ATP_Olambda__668,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_li(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu2),Uuc)) ).

% ATP.lambda_668
tff(fact_10779_ATP_Olambda__669,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ll(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc)) ).

% ATP.lambda_669
tff(fact_10780_ATP_Olambda__670,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_lj(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu2),Uuc)) ).

% ATP.lambda_670
tff(fact_10781_ATP_Olambda__671,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_lk(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu2),Uuc)) ).

% ATP.lambda_671
tff(fact_10782_ATP_Olambda__672,axiom,
    ! [Uu2: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_lb(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_672
tff(fact_10783_ATP_Olambda__673,axiom,
    ! [Uu2: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_lh(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_673
tff(fact_10784_ATP_Olambda__674,axiom,
    ! [Uu2: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ld(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu2),Uuc)),minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_674
tff(fact_10785_ATP_Olambda__675,axiom,
    ! [Uu2: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_lf(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_675
tff(fact_10786_ATP_Olambda__676,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_lm(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_676
tff(fact_10787_ATP_Olambda__677,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lp(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_677
tff(fact_10788_ATP_Olambda__678,axiom,
    ! [Uu2: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ln(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Uu2),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_678
tff(fact_10789_ATP_Olambda__679,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_lo(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_679
tff(fact_10790_ATP_Olambda__680,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_wp(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_680
tff(fact_10791_ATP_Olambda__681,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aac(Uu2),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,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_681
tff(fact_10792_ATP_Olambda__682,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abe(Uu2),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,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_682
tff(fact_10793_ATP_Olambda__683,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aaa(Uu2),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,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_683
tff(fact_10794_ATP_Olambda__684,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abc(Uu2),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,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_684
tff(fact_10795_ATP_Olambda__685,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abg(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_685
tff(fact_10796_ATP_Olambda__686,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abi(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_686
tff(fact_10797_ATP_Olambda__687,axiom,
    ! [Uu2: code_integer,Uua: $o,Uub: code_integer,Uuc: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aa_o_f2459543391104474489un_o_o(aTP_Lamm_zh(Uu2),(Uua)),Uub),(Uuc))
    <=> ( ( Uu2 = Uub )
        & ( (Uua)
        <=> (Uuc) ) ) ) ).

% ATP.lambda_687
tff(fact_10798_ATP_Olambda__688,axiom,
    ! [Uu2: num,Uua: num,Uub: num,Uuc: num] :
      ( aa_num_o(aa_num_fun_num_o(aa_num5226579579167915582_num_o(aTP_Lamm_zi(Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_688
tff(fact_10799_ATP_Olambda__689,axiom,
    ! [Uu2: nat,Uua: num,Uub: nat,Uuc: num] :
      ( aa_num_o(aa_nat_fun_num_o(aa_num3446304307616776968_num_o(aTP_Lamm_zj(Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_689
tff(fact_10800_ATP_Olambda__690,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_zk(Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_690
tff(fact_10801_ATP_Olambda__691,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_zl(Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_691
tff(fact_10802_ATP_Olambda__692,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dv(Uu2),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_692
tff(fact_10803_ATP_Olambda__693,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ed(Uu2),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_693
tff(fact_10804_ATP_Olambda__694,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_dr(Uu2),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu2)
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_694
tff(fact_10805_ATP_Olambda__695,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dz(Uu2),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_695
tff(fact_10806_ATP_Olambda__696,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_eh(Uu2),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_696
tff(fact_10807_ATP_Olambda__697,axiom,
    ! [Uu2: 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_dt(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_697
tff(fact_10808_ATP_Olambda__698,axiom,
    ! [Uu2: 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(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_698
tff(fact_10809_ATP_Olambda__699,axiom,
    ! [Uu2: 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_dp(Uu2),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu2)
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_699
tff(fact_10810_ATP_Olambda__700,axiom,
    ! [Uu2: 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_dx(Uu2),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_700
tff(fact_10811_ATP_Olambda__701,axiom,
    ! [Uu2: 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_ef(Uu2),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_701
tff(fact_10812_ATP_Olambda__702,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_ep(Uu2),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_702
tff(fact_10813_ATP_Olambda__703,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ex(Uu2),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_703
tff(fact_10814_ATP_Olambda__704,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_el(Uu2),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu2)
        & ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_704
tff(fact_10815_ATP_Olambda__705,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_et(Uu2),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_705
tff(fact_10816_ATP_Olambda__706,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_fb(Uu2),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_706
tff(fact_10817_ATP_Olambda__707,axiom,
    ! [Uu2: 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_en(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_707
tff(fact_10818_ATP_Olambda__708,axiom,
    ! [Uu2: 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_ev(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_708
tff(fact_10819_ATP_Olambda__709,axiom,
    ! [Uu2: 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_ej(Uu2),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu2)
        & ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_709
tff(fact_10820_ATP_Olambda__710,axiom,
    ! [Uu2: 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_er(Uu2),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_710
tff(fact_10821_ATP_Olambda__711,axiom,
    ! [Uu2: 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_ez(Uu2),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_711
tff(fact_10822_ATP_Olambda__712,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_me(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_712
tff(fact_10823_ATP_Olambda__713,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_mf(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_713
tff(fact_10824_ATP_Olambda__714,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vv(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_714
tff(fact_10825_ATP_Olambda__715,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wb(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_715
tff(fact_10826_ATP_Olambda__716,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_vx(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_716
tff(fact_10827_ATP_Olambda__717,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vz(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uuc)),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_717
tff(fact_10828_ATP_Olambda__718,axiom,
    ! [Uu2: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_tq(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_718
tff(fact_10829_ATP_Olambda__719,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_tt(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_719
tff(fact_10830_ATP_Olambda__720,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ub(Uu2),Uua),Uub),Uuc) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_720
tff(fact_10831_ATP_Olambda__721,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_uk(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_721
tff(fact_10832_ATP_Olambda__722,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_ud(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_722
tff(fact_10833_ATP_Olambda__723,axiom,
    ! [Uu2: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_fj(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu2,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_723
tff(fact_10834_ATP_Olambda__724,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_fn(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_724
tff(fact_10835_ATP_Olambda__725,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_fl(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_725
tff(fact_10836_ATP_Olambda__726,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_fh(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_726
tff(fact_10837_ATP_Olambda__727,axiom,
    ! [Uu2: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_jz((Uu2)),Uua),Uub),Uuc)
    <=> ( (Uu2)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_727
tff(fact_10838_ATP_Olambda__728,axiom,
    ! [Uu2: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_hk(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_728
tff(fact_10839_ATP_Olambda__729,axiom,
    ! [Uu2: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_hg(Uu2),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_729
tff(fact_10840_ATP_Olambda__730,axiom,
    ! [Uu2: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_hh(Uu2),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu2,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_730
tff(fact_10841_ATP_Olambda__731,axiom,
    ! [Uu2: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_hf(Uu2),Uua),Uub),Uuc)
    <=> aa_o_o(Uu2,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_731
tff(fact_10842_ATP_Olambda__732,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun7716529229553496594at_nat(aTP_Lamm_he(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_732
tff(fact_10843_ATP_Olambda__733,axiom,
    ! [Uu2: fun_na8713714802964096592at_nat,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aa_fun2484538639213223237at_nat(aTP_Lamm_hl(Uu2),Uua),Uub),Uuc) = aa_nat7997280740862404369at_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_733
tff(fact_10844_ATP_Olambda__734,axiom,
    ! [Uu2: fun_na3970208099681795815_nat_o,Uua: fun_nat_fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun2100511394839469148_nat_o(aTP_Lamm_hi(Uu2),Uua),Uub),Uuc) = aa_nat7695548938202568488_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_734
tff(fact_10845_ATP_Olambda__735,axiom,
    ! [Uu2: fun_fu1413608398870722896at_nat,Uua: fun_na1668100129766719005at_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun8817627472026143685at_nat(aTP_Lamm_hm(Uu2),Uua),Uub),Uuc) = aa_fun7582795104022253201at_nat(Uu2,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uua,Uub),Uuc)) ).

% ATP.lambda_735
tff(fact_10846_ATP_Olambda__736,axiom,
    ! [Uu2: fun_fu1115759055147092071_o_nat,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun3680403850923830236at_nat(aTP_Lamm_hj(Uu2),Uua),Uub),Uuc) = aa_fun1432809138467554728_o_nat(Uu2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_736
tff(fact_10847_ATP_Olambda__737,axiom,
    ! [Uu2: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_hn(Uu2),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_737
tff(fact_10848_ATP_Olambda__738,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ih(Uu2),Uua),Uub),Uuc) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_738
tff(fact_10849_ATP_Olambda__739,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ig(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_739
tff(fact_10850_ATP_Olambda__740,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_rv(Uu2),Uua),Uub),Uuc) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_740
tff(fact_10851_ATP_Olambda__741,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aae(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_741
tff(fact_10852_ATP_Olambda__742,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aag(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_742
tff(fact_10853_ATP_Olambda__743,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aak(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_743
tff(fact_10854_ATP_Olambda__744,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aai(Uu2),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_744
tff(fact_10855_ATP_Olambda__745,axiom,
    ! [Uu2: fun_real_o,Uua: fun_complex_o,Uub: fun_re7208826350674946562omplex,Uuc: complex] :
      ( aa_complex_o(aa_fun434462017428657539plex_o(aa_fun7128336235388974163plex_o(aTP_Lamm_dj(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: complex] :
          ( ( Uuc = aa_complex_complex(aa_rea2618301233319580611omplex(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_complex_o(Uua,Y5) ) ) ).

% ATP.lambda_745
tff(fact_10856_ATP_Olambda__746,axiom,
    ! [Uu2: fun_real_o,Uua: fun_complex_o,Uub: fun_re2354894731504631552x_real,Uuc: real] :
      ( aa_real_o(aa_fun2419829551541144455real_o(aa_fun4137116173344588375real_o(aTP_Lamm_dh(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: complex] :
          ( ( Uuc = aa_complex_real(aa_rea4562580341872191937x_real(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_complex_o(Uua,Y5) ) ) ).

% ATP.lambda_746
tff(fact_10857_ATP_Olambda__747,axiom,
    ! [Uu2: fun_real_o,Uua: fun_complex_o,Uub: fun_re175488254389786404ex_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun6402932185680802879_nat_o(aa_fun3831867060791545871_nat_o(aTP_Lamm_di(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: complex] :
          ( ( Uuc = aa_complex_nat(aa_rea7691190107667057125ex_nat(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_complex_o(Uua,Y5) ) ) ).

% ATP.lambda_747
tff(fact_10858_ATP_Olambda__748,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_o,Uub: fun_re7812779276277946752omplex,Uuc: complex] :
      ( aa_complex_o(aa_fun890640625120065285plex_o(aa_fun6537733294925973843plex_o(aTP_Lamm_dd(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: real] :
          ( ( Uuc = aa_real_complex(aa_rea2649881322057434177omplex(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_748
tff(fact_10859_ATP_Olambda__749,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_o,Uub: fun_re7681423839302404990l_real,Uuc: real] :
      ( aa_real_o(aa_fun6833002213068288777real_o(aa_fun8577581014811940695real_o(aTP_Lamm_db(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: real] :
          ( ( Uuc = aa_real_real(aa_rea7340605660300375871l_real(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_749
tff(fact_10860_ATP_Olambda__750,axiom,
    ! [Uu2: fun_real_o,Uua: fun_real_o,Uub: fun_re7428432522468557474al_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun6822315102276328897_nat_o(aa_fun4096426214187263247_nat_o(aTP_Lamm_dc(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: real] :
          ( ( Uuc = aa_real_nat(aa_real_fun_real_nat(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_750
tff(fact_10861_ATP_Olambda__751,axiom,
    ! [Uu2: fun_real_o,Uua: fun_nat_o,Uub: fun_re2084383175013312036omplex,Uuc: complex] :
      ( aa_complex_o(aa_fun2810168320254036705plex_o(aa_fun467192419720471251plex_o(aTP_Lamm_dg(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: nat] :
          ( ( Uuc = aa_nat_complex(aa_rea1531280491672885477omplex(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_nat_o(Uua,Y5) ) ) ).

% ATP.lambda_751
tff(fact_10862_ATP_Olambda__752,axiom,
    ! [Uu2: fun_real_o,Uua: fun_nat_o,Uub: fun_re4238438383274954274t_real,Uuc: real] :
      ( aa_real_o(aa_fun3910182461874428645real_o(aa_fun8527292551407841495real_o(aTP_Lamm_de(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: nat] :
          ( ( Uuc = aa_nat_real(aa_real_fun_nat_real(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_nat_o(Uua,Y5) ) ) ).

% ATP.lambda_752
tff(fact_10863_ATP_Olambda__753,axiom,
    ! [Uu2: fun_real_o,Uua: fun_nat_o,Uub: fun_real_fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun1675651493036827037_nat_o(aa_fun957765429596782223_nat_o(aTP_Lamm_df(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: real,Y5: nat] :
          ( ( Uuc = aa_nat_nat(aa_real_fun_nat_nat(Uub,X3),Y5) )
          & aa_real_o(Uu2,X3)
          & aa_nat_o(Uua,Y5) ) ) ).

% ATP.lambda_753
tff(fact_10864_ATP_Olambda__754,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_real_o,Uub: fun_na6724076748692969250l_real,Uuc: real] :
      ( aa_real_o(aa_fun1426617528609952229real_o(aa_fun7503860880236520883real_o(aTP_Lamm_dk(Uu2),Uua),Uub),Uuc)
    <=> ? [X3: nat,Y5: real] :
          ( ( Uuc = aa_real_real(aa_nat_fun_real_real(Uub,X3),Y5) )
          & aa_nat_o(Uu2,X3)
          & aa_real_o(Uua,Y5) ) ) ).

% ATP.lambda_754
tff(fact_10865_ATP_Olambda__755,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_vu(Uu2),Uua),Uub),Uuc),Uud) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_vt(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_755
tff(fact_10866_ATP_Olambda__756,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_vs(Uu2),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_vr(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_756
tff(fact_10867_ATP_Olambda__757,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_vq(Uu2),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_vp(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_757
tff(fact_10868_ATP_Olambda__758,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_vo(Uu2),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_vn(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_758
tff(fact_10869_ATP_Olambda__759,axiom,
    ! [Uu2: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_acy(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_acx(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))))) ).

% ATP.lambda_759
tff(fact_10870_ATP_Olambda__760,axiom,
    ! [Uu2: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_wc(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_wb(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud)) ).

% ATP.lambda_760
tff(fact_10871_ATP_Olambda__761,axiom,
    ! [Uu2: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_wa(Uu2),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vz(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_761
tff(fact_10872_ATP_Olambda__762,axiom,
    ! [Uu2: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_vy(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_vx(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_762
tff(fact_10873_ATP_Olambda__763,axiom,
    ! [Uu2: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_vw(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vv(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu2))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_763
tff(fact_10874_ATP_Olambda__764,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_vk(Uu2),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,Uu2),Uud))) ).

% ATP.lambda_764
tff(fact_10875_ATP_Olambda__765,axiom,
    ! [Uu2: 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_vm(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(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,Uu2),Uud))) ).

% ATP.lambda_765
tff(fact_10876_ATP_Olambda__766,axiom,
    ! [Uu2: 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_vl(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_766
tff(fact_10877_ATP_Olambda__767,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_uw(Uu2),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,Uu2)),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,Uu2),Uud))) ).

% ATP.lambda_767
tff(fact_10878_ATP_Olambda__768,axiom,
    ! [Uu2: 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_va(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu2)),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,Uu2),Uud))) ).

% ATP.lambda_768
tff(fact_10879_ATP_Olambda__769,axiom,
    ! [Uu2: 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_uy(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Uu2)),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,Uu2),Uud))) ).

% ATP.lambda_769
tff(fact_10880_ATP_Olambda__770,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ux(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_770
tff(fact_10881_ATP_Olambda__771,axiom,
    ! [Uu2: 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_vb(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_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,Uu2),Uud))) ).

% ATP.lambda_771
tff(fact_10882_ATP_Olambda__772,axiom,
    ! [Uu2: 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_uz(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(uminus_uminus_rat(Uua)),Uud)),aa_nat_rat(power_power_rat(uminus_uminus_rat(Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uud))) ).

% ATP.lambda_772
tff(fact_10883_ATP_Olambda__773,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_vn(Uu2),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_773
tff(fact_10884_ATP_Olambda__774,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: real,Uuc: nat,Uud: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_vt(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_774
tff(fact_10885_ATP_Olambda__775,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: rat,Uuc: nat,Uud: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_vp(Uu2),Uua),Uub),Uuc),Uud) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_775
tff(fact_10886_ATP_Olambda__776,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: int,Uuc: nat,Uud: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_vr(Uu2),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_776
tff(fact_10887_ATP_Olambda__777,axiom,
    ! [Uu2: 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_abl(Uu2),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))),Uu2))
        & ! [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,
            ! [X3: vEBT_VEBT] :
              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Uua))
             => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X7) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X3)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_777
tff(fact_10888_ATP_Olambda__778,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ka,Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_778
tff(fact_10889_ATP_Olambda__779,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_ak,Uu2) = Uu2 ).

% ATP.lambda_779
tff(fact_10890_ATP_Olambda__780,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_al,Uu2) = Uu2 ).

% ATP.lambda_780
tff(fact_10891_ATP_Olambda__781,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_am,Uu2) = Uu2 ).

% ATP.lambda_781
tff(fact_10892_ATP_Olambda__782,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_an,Uu2) = Uu2 ).

% ATP.lambda_782
tff(fact_10893_ATP_Olambda__783,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_ao,Uu2) = Uu2 ).

% ATP.lambda_783
tff(fact_10894_ATP_Olambda__784,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_be,Uu2) = zero_zero_complex ).

% ATP.lambda_784
tff(fact_10895_ATP_Olambda__785,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_bf,Uu2) = zero_zero_real ).

% ATP.lambda_785
tff(fact_10896_ATP_Olambda__786,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_bg,Uu2) = zero_zero_rat ).

% ATP.lambda_786
tff(fact_10897_ATP_Olambda__787,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_bh,Uu2) = zero_zero_nat ).

% ATP.lambda_787
tff(fact_10898_ATP_Olambda__788,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_bi,Uu2) = zero_zero_int ).

% ATP.lambda_788
tff(fact_10899_ATP_Olambda__789,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_os,Uu2) = one_one_nat ).

% ATP.lambda_789
tff(fact_10900_ATP_Olambda__790,axiom,
    ! [Uu2: nat] : aa_nat_int(aTP_Lamm_ot,Uu2) = one_one_int ).

% ATP.lambda_790
tff(fact_10901_ATP_Olambda__791,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_ou,Uu2) = one_one_int ).

% ATP.lambda_791
tff(fact_10902_ATP_Olambda__792,axiom,
    ! [Uu2: nat] :
      ( aa_nat_o(aTP_Lamm_aas,Uu2)
    <=> $false ) ).

% ATP.lambda_792
tff(fact_10903_ATP_Olambda__793,axiom,
    ! [Uu2: nat] :
      ( aa_nat_o(aTP_Lamm_aar,Uu2)
    <=> $true ) ).

% ATP.lambda_793

% Conjectures (1)
tff(conj_0,conjecture,
    ( ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = some_nat(sxa) )
  <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary))),xa),sxa) ) ).

%------------------------------------------------------------------------------