%------------------------------------------------------------------------------
% File     : ITP259_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Delete 01385_095048
% 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    : 0072_VEBT_Delete_01385_095048 [Des22]

% Status   : Theorem
% Rating   : 0.89 v8.2.0, 1.00 v8.1.0
% Syntax   : Number of formulae    : 13420 (3938 unt;2515 typ;   0 def)
%            Number of atoms       : 27737 (10191 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 19625 (2793   ~; 541   |;1920   &)
%                                         (2713 <=>;11658  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of types       :  429 ( 428 usr)
%            Number of type conns  : 2428 (1621   >; 807   *;   0   +;   0  <<)
%            Number of predicates  :  114 ( 111 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1976 (1976 usr; 466 con; 0-8 aty)
%            Number of variables   : 26519 (25793   !; 726   ?;26519   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 06:36:48.226
%------------------------------------------------------------------------------
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    aTP_Lamm_a: set_complex > fun_complex_bool ).

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

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: set_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: real > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: nat > fun_na5042103099923536755t_bool ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: extended_enat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: extended_enat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: real > fun_re8658731967611918585t_real ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: nat > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: nat > fun_na8797796813480665199d_enat ).

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

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: set_nat > fun_nat_bool ).

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

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

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

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: vEBT_VEBT > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: vEBT_VEBT > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_real_real ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_int_int ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_real_real ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_int_int ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_nat_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: rat > fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_Extended_enat > fun_na4695037785009495611t_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_int > fun_na7919044170365968317t_bool ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_complex > fun_na376774557589183419x_bool ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_Pr1261947904930325089at_nat > fun_na5265545386895126880t_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_Extended_enat > fun_na4695037785009495611t_bool ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_VEBT_VEBT > fun_na2508794161012263405T_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_bool > fun_na205931550456536010l_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: nat > fun_co1378794544173401837x_bool ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: nat > fun_complex_bool ).

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

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

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

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_nat_real ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_nat_bool > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_nat_bool > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_nat_bool > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: set_nat > fun_fu8464300013002972271at_int ).

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

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_int > fun_fu8464300013002972271at_int ).

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

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

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_complex > fun_co8378349014452122532t_real ).

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

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_real > fun_fu914841811607101702l_bool ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_real > fun_fu7100649987335273288l_bool ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_nat > fun_fu4346542433904371390t_bool ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_nat > fun_fu8382797250367883996t_bool ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_int > fun_fu7152481525987436038t_bool ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_int > fun_fu5129045444369776328t_bool ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_complex > fun_fu5519326811792979714x_bool ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_complex > fun_fu3191020621175015362x_bool ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_Extended_enat > fun_fu7727529105362479790t_bool ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_Extended_enat > fun_fu4596718790507443722t_bool ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_Extended_enat > fun_fu3738011059562840620t_bool ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_Extended_enat > fun_fu5527893311006036234t_bool ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_real > fun_fu2568719150003003596l_bool ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_nat > fun_fu7195134416904984768t_bool ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_nat > fun_fu8639250699203096928t_bool ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_int > fun_fu549321512867696136t_bool ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_int > fun_fu3401574108658714444t_bool ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_complex > fun_fu6356453956411242500x_bool ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_complex > fun_fu5995210064384465734x_bool ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_Extended_enat > fun_fu3738011059562840620t_bool ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_Extended_enat > fun_fu5527893311006036234t_bool ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_real > fun_fu2277326744902863452l_bool ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_real > fun_fu6750725090257165812l_bool ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_nat > fun_fu3052598110916186900t_bool ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_nat > fun_fu2971308621445215880t_bool ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_int > fun_fu337402001079966300t_bool ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_int > fun_fu163127709195666036t_bool ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_complex > fun_fu3810065756161129304x_bool ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_complex > fun_fu9216119399439645550x_bool ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_Extended_enat > fun_fu8394451710700313048t_bool ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_Extended_enat > fun_fu5057839232988439434t_bool ).

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

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_complex_complex ).

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

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_int_nat > fun_int_int ).

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

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

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: extended_enat > fun_fu263853427677366787at_rat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: extended_enat > fun_fu263853427677366787at_rat ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: real > fun_fu3214020956574444655al_nat ).

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

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: set_real > fun_fu5874800227564645059l_bool ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_int > fun_fu7822069302422884163t_bool ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: set_complex > fun_fu5204410621998794179x_bool ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_Extended_enat > fun_fu739766798173130341t_bool ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_Ex1451620480498224165t_real > fun_fu4696490002264930058t_real ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_real_rat > fun_fu7249360421009165912al_rat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_nat > fun_fu3889889255038636099t_bool ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_rat > fun_fu2739651132436891864at_rat ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_int_rat > fun_fu2671659548940339672nt_rat ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_complex_rat > fun_fu4947971316162376536ex_rat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_Ex4152069839811072081at_rat > fun_fu8357369079510409782at_rat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_real_nat > fun_fu4405974721632067664al_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_Ex1451620480498224165t_real > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_complex_rat > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_Ex4152069839811072081at_rat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_real_nat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_int_nat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: nat > fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_nat_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_fun_nat_bool ).

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

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

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

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

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: int > fun_int_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_complex > fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: real > fun_fu4063443379060642216t_real ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_real > fun_nat_fun_nat_real ).

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

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

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

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: complex > fun_co6883864892131739507omplex ).

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

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: rat > fun_ra5513060666784058567at_rat ).

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

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: int > fun_in6793278964266839919at_int ).

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

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: rat > fun_nat_fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_real > fun_fu6899955116716589935t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: rat > fun_nat_fun_rat_rat ).

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

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

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_real_real ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_int_int ).

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

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

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

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_nat_int ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_int_int ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: extended_enat > fun_fu6052821276749137539omplex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: extended_enat > fun_fu6052821276749137539omplex ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: extended_enat > fun_fu7336479014630192387t_real ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_real_complex > fun_fu1393723446858488622omplex ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_complex > fun_fu7905158015251895726omplex ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_int_complex > fun_fu6363802315473983150omplex ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_complex_complex > fun_fu248857536038387758omplex ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_Ex3741802288643809319omplex > fun_fu5517063109087654668omplex ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_real_real > fun_fu1760894646116856364l_real ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_nat_real > fun_fu9152424228179946668t_real ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_int_real > fun_fu6185597869953674668t_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_complex_real > fun_fu1911030855436031788x_real ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_Ex1451620480498224165t_real > fun_fu4696490002264930058t_real ).

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

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

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

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

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_Ex3741802288643809319omplex > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_int_real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_complex_real > fun_complex_bool ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_Ex1451620480498224165t_real > fun_Ex8563670519792538868t_bool ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_real_rat > fun_real_bool ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_int_rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_int > fun_na7654559027617327379at_int ).

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

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

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

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: complex > fun_na1591161823130732966omplex ).

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

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: complex > fun_na1142814060743658952omplex ).

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

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: rat > fun_nat_fun_nat_rat ).

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: set_nat > fun_set_nat_bool ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_rat_fun_rat_bool ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_num_fun_num_bool ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_fun_nat_nat ).

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_real > fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_set_nat_real > fun_set_nat_real ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_Pr672294278902479040t_real > fun_Pr672294278902479040t_real ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_Ex1451620480498224165t_real > fun_Ex1451620480498224165t_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: rat > fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_real > fun_na6801867027684862029l_bool ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: nat > fun_re8542384036533109323t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_nat_fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_int > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_int_int > fun_int_complex ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: int > fun_int_fun_int_bool ).

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

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_real_bool ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_real_bool ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: int > fun_in6148020733644367646t_bool ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: rat > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: rat > fun_int_bool ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: real > fun_int_bool ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: complex > fun_na5941084329111710283x_bool ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_na1668100129766719005at_nat ).

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: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: fun_int_nat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: int > fun_int_fun_int_bool ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: nat > fun_nat_fun_nat_bool ).

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

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_real_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_real_real ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_nat_real ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: real > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_real_bool > fun_re1255401033344146673l_bool ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_real_real > fun_real_bool ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_real_real > fun_fu7376312551830202376l_bool ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_real_bool > fun_real_bool ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_real_real ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: nat > fun_li8232715659020003717t_bool ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_real_bool ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_real_bool ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: complex > fun_real_bool ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: rat > fun_Pr803113274115125473t_bool ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: nat > fun_na5614638261385347865t_bool ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: nat > fun_na5614638261385347865t_bool ).

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

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__zq____,type,
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tff(sy_c_ATP_058Lamm__zr____,type,
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tff(sy_c_ATP_058Lamm__zs____,type,
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tff(sy_c_ATP_058Lamm__zt____,type,
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tff(sy_c_ATP_058Lamm__zu____,type,
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tff(sy_c_ATP_058Lamm__zv____,type,
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tff(sy_c_ATP_058Lamm__zw____,type,
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tff(sy_c_ATP_058Lamm__zx____,type,
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tff(sy_c_ATP_058Lamm__zy____,type,
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tff(sy_c_ATP_058Lamm__zz____,type,
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tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
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tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__HOL__Obool_001t__HOL__Obool,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Oand__int__rel,type,
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tff(sy_c_Bit__Operations_Oand__not__num,type,
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tff(sy_c_List_Oappend_001t__Nat__Onat,type,
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tff(sy_c_List_Ocount__list_001t__Int__Oint,type,
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tff(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Odrop_001t__Nat__Onat,type,
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tff(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olast_001t__Nat__Onat,type,
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tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__HOL__Obool,type,
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tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
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tff(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: fun_nat_list_nat ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun_nat_fun_nat_bool ).

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_li8598321315282354825t_bool ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun_na1476630412504255114at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr248683899324676269t_bool ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    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: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
    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: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

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

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: 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: fun_num_rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: fun_num_real ).

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

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Num_Osqr,type,
    sqr: fun_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,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001t__HOL__Obool_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o7912701867834613557at_nat: ( bool * fun_Pr7757163693017731481t_bool * option4927543243414619207at_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_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_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > fun_nat_set_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__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

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__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

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_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: fun_nat_bool > nat ).

tff(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
    ord_Least_real: fun_real_bool > real ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: fun_Co8310303586393106575r_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > fun_Ex8563670519792538868t_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > fun_set_complex_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le2529575680413868914d_enat: set_Extended_enat > fun_se2015444840911768724t_bool ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > fun_set_int_bool ).

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

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,
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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_001t__fun_It__Complex__Ocomplex_Mt__HOL__Obool_J,type,
    ord_le3776645642988763374x_bool: ( fun_complex_bool * fun_complex_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_Mt__HOL__Obool_J,type,
    ord_le3716944864119727344t_bool: ( fun_int_bool * fun_int_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    ord_le6841235830588351564t_bool: ( fun_nat_bool * fun_nat_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    ord_le7519368973566061808l_bool: ( fun_real_bool * fun_real_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__HOL__Obool_J,type,
    ord_le4523237222650602006t_bool: ( fun_set_nat_bool * fun_set_nat_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_re1255401033344146673l_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > fun_set_complex_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le7203529160286727270d_enat: set_Extended_enat > fun_se2015444840911768724t_bool ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__HOL__Obool_J,type,
    ord_less_eq_set_bool: ( set_bool * set_bool ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > fun_set_int_bool ).

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_bool ).

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__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > fun_se8074476271188067385t_bool ).

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

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

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

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

tff(sy_c_Orderings_Oord__class_Omax_001t__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,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_ma4205026669011143323d_enat: ( set_Extended_enat * set_Extended_enat ) > set_Extended_enat ).

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

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

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: extended_enat > fun_Ex5852502975177633731d_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: fun_int_fun_int_int ).

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_bool > 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_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: fun_real_real > $o ).

tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( fun_nat_fun_nat_bool * fun_nat_fun_nat_bool ) > fun_nat_bool ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__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,
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tff(sy_c_Set_OPow_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
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tff(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
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tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

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tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
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tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
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tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Oarcsin,type,
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tff(sy_c_Transcendental_Oarctan,type,
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tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Ocos__coeff,type,
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tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
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tff(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
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tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
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tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Opowr__real,type,
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tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Osin__coeff,type,
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tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
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tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
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tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
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tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
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tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
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tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
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tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
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    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_bool ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
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tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
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tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
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tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
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tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
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tff(sy_c_VEBT__Definitions_Oset__vebt,type,
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tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
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tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
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tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
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tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE5877677516038742509T_bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun_nat_bool ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option_nat * option_nat ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option_nat * option_nat ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option_nat * option_nat ) > $o ).

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

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

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,
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tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
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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_VE5877677516038742509T_bool ).

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

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
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tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
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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_Pr817965127378462947t_bool ).

tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: fun_li8598321315282354825t_bool > fun_list_nat_bool ).

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tff(sy_c_Wellfounded_Oless__than,type,
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tff(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
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    aa_fun3352386287069783311l_bool: ( fun_fu7511176317168184086l_bool * fun_real_bool ) > bool ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    aa_fun6813637976755515871omplex: ( fun_fu1393723446858488622omplex * fun_real_bool ) > fun_real_complex ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    aa_fun6476784834714468668l_bool: ( fun_fu5874800227564645059l_bool * fun_real_bool ) > fun_real_bool ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
    aa_fun1504556115071737857al_nat: ( fun_fu4405974721632067664al_nat * fun_real_bool ) > fun_real_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
    aa_fun7219699593533116937al_rat: ( fun_fu7249360421009165912al_rat * fun_real_bool ) > fun_real_rat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun6348708079608492509l_real: ( fun_fu1760894646116856364l_real * fun_real_bool ) > fun_real_real ).

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

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
    aa_fun8987621170725249221l_bool: ( fun_fu2277326744902863452l_bool * fun_real_rat ) > fun_real_bool ).

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

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Rat__Orat_J_Mt__fun_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
    aa_fun8530242357055030877l_bool: ( fun_fu6750725090257165812l_bool * fun_real_rat ) > fun_fu2277326744902863452l_bool ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__HOL__Obool,type,
    aa_fun9137951878280015556l_bool: ( fun_fu7307090382573344987l_bool * fun_real_real ) > bool ).

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_fEx_001t__Nat__Onat,type,
    fEx_nat: fun_fu7168253649841883962l_bool ).

tff(sy_c_fEx_001t__Real__Oreal,type,
    fEx_real: fun_fu7511176317168184086l_bool ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun_bool_bool ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: bool > fun_bool_bool ).

tff(sy_c_fdisj,type,
    fdisj: bool > fun_bool_bool ).

tff(sy_c_fequal_001t__Code____Numeral__Ointeger,type,
    fequal_Code_integer: fun_Co8310303586393106575r_bool ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: fun_co1378794544173401837x_bool ).

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

tff(sy_c_fequal_001t__HOL__Obool,type,
    fequal_bool: fun_bo1841576699660343469l_bool ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_bool ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_bool ).

tff(sy_c_fequal_001t__Option__Ooption_It__Nat__Onat_J,type,
    fequal_option_nat: ( option_nat * option_nat ) > bool ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_re1255401033344146673l_bool ).

tff(sy_c_fimplies,type,
    fimplies: fun_bo1841576699660343469l_bool ).

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

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

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

tff(sy_c_member_001t__HOL__Obool,type,
    member_bool: ( bool * set_bool ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: int > fun_set_int_bool ).

tff(sy_c_member_001t__List__Olist_It__HOL__Obool_J,type,
    member_list_bool: ( list_bool * set_list_bool ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_bool ).

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > fun_se8074476271188067385t_bool ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: ( produc859450856879609959at_nat * set_Pr8693737435421807431at_nat ) > $o ).

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

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * 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: fun_VE7346385682144900493T_bool ).

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

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10838)
tff(fact_0__C5_OIH_C_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(summary,X)),Y))
    <=> ( ( X != Y )
        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),Y)) ) ) ).

% "5.IH"(2)
tff(fact_1_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1)) ) ).

% not_min_Null_member
tff(fact_2__C5_Ohyps_C_I3_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "5.hyps"(3)
tff(fact_3__C5_Ohyps_C_I1_J,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).

% "5.hyps"(1)
tff(fact_4__C5_Ohyps_C_I7_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).

% "5.hyps"(7)
tff(fact_5__C5_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ).

% "5.hyps"(6)
tff(fact_6_mimapr,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),ma)) ).

% mimapr
tff(fact_7_bit__split__inv,axiom,
    ! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).

% bit_split_inv
tff(fact_8__C5_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "5.hyps"(4)
tff(fact_9_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
    <=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(T),X))
        | pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).

% both_member_options_def
tff(fact_10_False,axiom,
    xa = mi ).

% False
tff(fact_11__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),xa))
    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,xa),ma)) ) ).

% \<open>mi \<le> x \<and> x \<le> ma\<close>
tff(fact_12_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_13_max__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),X)) ) ) ) ).

% max_in_set_def
tff(fact_14_min__in__set__def,axiom,
    ! [Xs: set_nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(Xs),X))
    <=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X3)) ) ) ) ).

% min_in_set_def
tff(fact_15__C5_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(treeList)))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(X2),na))
        & ! [Xa: nat,Xb: nat] :
            ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(X2,Xa)),Xb))
          <=> ( ( Xa != Xb )
              & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),Xb)) ) ) ) ) ).

% "5.IH"(1)
tff(fact_16__092_060open_062x_A_061_Ami_092_060close_062,axiom,
    xa = mi ).

% \<open>x = mi\<close>
tff(fact_17_nmpr,axiom,
    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),na))
    & ( m = aa_nat_nat(suc,na) ) ) ).

% nmpr
tff(fact_18_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% nat_add_left_cancel_le
tff(fact_19_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% nat_add_left_cancel_less
tff(fact_20_add__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% add_Suc_right
tff(fact_21_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_nat_nat(suc,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M)) ) ).

% Suc_le_mono
tff(fact_22_lessI,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(suc,N2))) ).

% lessI
tff(fact_23_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) ) ).

% Suc_mono
tff(fact_24_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% Suc_less_eq
tff(fact_25_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_left
tff(fact_26_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_left
tff(fact_27_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_left
tff(fact_28_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_left
tff(fact_29_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_cancel_right
tff(fact_30_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_cancel_right
tff(fact_31_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_cancel_right
tff(fact_32_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_cancel_right
tff(fact_33_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_left
tff(fact_34_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_left
tff(fact_35_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_left
tff(fact_36_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_left
tff(fact_37_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_cancel_right
tff(fact_38_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_cancel_right
tff(fact_39_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_cancel_right
tff(fact_40_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_cancel_right
tff(fact_41_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_42_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_43_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_44_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_45_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_46_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_47_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_48_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_49_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_50_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_51_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_52_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_53_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_54_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_55_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_56_Suc__eq__plus1__left,axiom,
    ! [N2: nat] : aa_nat_nat(suc,N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N2) ).

% Suc_eq_plus1_left
tff(fact_57_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_58_Suc__eq__plus1,axiom,
    ! [N2: nat] : aa_nat_nat(suc,N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat) ).

% Suc_eq_plus1
tff(fact_59_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_60_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_61_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_62_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_63_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_64_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_65_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_66_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_67_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_68_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_69_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_70_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_71_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_72_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_73_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_74_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_75_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(member_complex(A),collect_complex(P)))
    <=> pp(aa_complex_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_76_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_bool] :
      ( member_real(A,collect_real(P))
    <=> pp(aa_real_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_77_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_bool] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> pp(aa_list_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_78_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_bool] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> pp(aa_set_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_79_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
    <=> pp(aa_nat_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_80_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(A),collect_int(P)))
    <=> pp(aa_int_bool(P,A)) ) ).

% mem_Collect_eq
tff(fact_81_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_82_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_83_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_84_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_85_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_86_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_87_Collect__cong,axiom,
    ! [P: fun_real_bool,Q: fun_real_bool] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(Q,X4)) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_88_Collect__cong,axiom,
    ! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
      ( ! [X4: list_nat] :
          ( pp(aa_list_nat_bool(P,X4))
        <=> pp(aa_list_nat_bool(Q,X4)) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_89_Collect__cong,axiom,
    ! [P: fun_set_nat_bool,Q: fun_set_nat_bool] :
      ( ! [X4: set_nat] :
          ( pp(aa_set_nat_bool(P,X4))
        <=> pp(aa_set_nat_bool(Q,X4)) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_90_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(P,X4))
        <=> pp(aa_nat_bool(Q,X4)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_91_Collect__cong,axiom,
    ! [P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(Q,X4)) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_92_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_93_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_94_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_95_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_96_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_97_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_98_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_99_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_100_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_101_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_102_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_103_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_104_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_105_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_106_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_107_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_108_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_109_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_110_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_111_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_112_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_113_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_114_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_115_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_116_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_117_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_118_n__not__Suc__n,axiom,
    ! [N2: nat] : N2 != aa_nat_nat(suc,N2) ).

% n_not_Suc_n
tff(fact_119_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_120_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_121_infinite__descent,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( ! [N: nat] :
          ( ~ pp(aa_nat_bool(P,N))
         => ? [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
              & ~ pp(aa_nat_bool(P,M2)) ) )
     => pp(aa_nat_bool(P,N2)) ) ).

% infinite_descent
tff(fact_122_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N)) )
     => pp(aa_nat_bool(P,N2)) ) ).

% nat_less_induct
tff(fact_123_less__irrefl__nat,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N2)) ).

% less_irrefl_nat
tff(fact_124_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_125_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
     => ( M != N2 ) ) ).

% less_not_refl2
tff(fact_126_less__not__refl,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N2)) ).

% less_not_refl
tff(fact_127_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ).

% nat_neq_iff
tff(fact_128_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => ? [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
            & ! [Y4: nat] :
                ( pp(aa_nat_bool(P,Y4))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y4),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_129_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M)) ) ).

% nat_le_linear
tff(fact_130_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( M = N2 ) ) ) ).

% le_antisym
tff(fact_131_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% eq_imp_le
tff(fact_132_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).

% le_trans
tff(fact_133_le__refl,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N2)) ).

% le_refl
tff(fact_134_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_right
tff(fact_135_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_136_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_right
tff(fact_137_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_right
tff(fact_138_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% add_le_imp_le_left
tff(fact_139_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_140_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% add_le_imp_le_left
tff(fact_141_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% add_le_imp_le_left
tff(fact_142_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ? [C2: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% le_iff_add
tff(fact_143_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_right_mono
tff(fact_144_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))) ) ).

% add_right_mono
tff(fact_145_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_right_mono
tff(fact_146_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_right_mono
tff(fact_147_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_148_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_left_mono
tff(fact_149_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% add_left_mono
tff(fact_150_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_left_mono
tff(fact_151_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_left_mono
tff(fact_152_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_mono
tff(fact_153_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_mono
tff(fact_154_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_mono
tff(fact_155_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_mono
tff(fact_156_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_157_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_158_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_159_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_160_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_161_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_162_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_163_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_164_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_165_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_166_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_167_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_168_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_right
tff(fact_169_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_170_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_right
tff(fact_171_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_right
tff(fact_172_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% add_less_imp_less_left
tff(fact_173_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_174_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% add_less_imp_less_left
tff(fact_175_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% add_less_imp_less_left
tff(fact_176_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).

% add_strict_right_mono
tff(fact_177_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))) ) ).

% add_strict_right_mono
tff(fact_178_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).

% add_strict_right_mono
tff(fact_179_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).

% add_strict_right_mono
tff(fact_180_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% add_strict_left_mono
tff(fact_181_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% add_strict_left_mono
tff(fact_182_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).

% add_strict_left_mono
tff(fact_183_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% add_strict_left_mono
tff(fact_184_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_strict_mono
tff(fact_185_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_strict_mono
tff(fact_186_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_strict_mono
tff(fact_187_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_strict_mono
tff(fact_188_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & ( K = L ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_189_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & ( K = L ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_190_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_191_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_192_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_193_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_194_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_195_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_196_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_197_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_198_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_199_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_200_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(suc,M)))
      <=> ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_201_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => pp(aa_nat_bool(P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I2)))
               => pp(aa_nat_bool(P,I2)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_202_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(suc,I2)))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J2))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),K2)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_203_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_204_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% Suc_less_SucD
tff(fact_205_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(suc,M)))
       => ( M = N2 ) ) ) ).

% less_antisym
tff(fact_206_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
    <=> ? [M3: nat] :
          ( ( M = aa_nat_nat(suc,M3) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M3)) ) ) ).

% Suc_less_eq2
tff(fact_207_All__less__Suc,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N2)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N2))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N2))
           => pp(aa_nat_bool(P,I3)) ) ) ) ).

% All_less_Suc
tff(fact_208_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(suc,M))) ) ).

% not_less_eq
tff(fact_209_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
        | ( M = N2 ) ) ) ).

% less_Suc_eq
tff(fact_210_Ex__less__Suc,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N2)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,N2))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N2))
            & pp(aa_nat_bool(P,I3)) ) ) ) ).

% Ex_less_Suc
tff(fact_211_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N2))) ) ).

% less_SucI
tff(fact_212_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N2)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => ( M = N2 ) ) ) ).

% less_SucE
tff(fact_213_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( ( aa_nat_nat(suc,M) != N2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N2)) ) ) ).

% Suc_lessI
tff(fact_214_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_215_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% Suc_lessD
tff(fact_216_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_217_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),X4))
       => ( ! [X4: nat,Y3: nat,Z: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Y3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y3),Z))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X4),Z)) ) )
         => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R,N),aa_nat_nat(suc,N)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,M),N2)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_218_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( pp(aa_nat_bool(P,M))
       => ( ! [N: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
             => ( pp(aa_nat_bool(P,N))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N))) ) )
         => pp(aa_nat_bool(P,N2)) ) ) ) ).

% nat_induct_at_least
tff(fact_219_full__nat__induct,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
             => pp(aa_nat_bool(P,M2)) )
         => pp(aa_nat_bool(P,N)) )
     => pp(aa_nat_bool(P,N2)) ) ).

% full_nat_induct
tff(fact_220_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N2)),M)) ) ).

% not_less_eq_eq
tff(fact_221_Suc__n__not__le__n,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N2)),N2)) ).

% Suc_n_not_le_n
tff(fact_222_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
        | ( M = aa_nat_nat(suc,N2) ) ) ) ).

% le_Suc_eq
tff(fact_223_Suc__le__D,axiom,
    ! [N2: nat,M4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N2)),M4))
     => ? [M5: nat] : M4 = aa_nat_nat(suc,M5) ) ).

% Suc_le_D
tff(fact_224_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2))) ) ).

% le_SucI
tff(fact_225_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( M = aa_nat_nat(suc,N2) ) ) ) ).

% le_SucE
tff(fact_226_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% Suc_leD
tff(fact_227_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N2)) ).

% add_Suc_shift
tff(fact_228_add__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% add_Suc
tff(fact_229_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_230_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_231_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( ( M != N2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% le_neq_implies_less
tff(fact_232_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
        | ( M = N2 ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% less_or_eq_imp_le
tff(fact_233_le__eq__less__or__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
        | ( M = N2 ) ) ) ).

% le_eq_less_or_eq
tff(fact_234_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% less_imp_le_nat
tff(fact_235_nat__less__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
        & ( M != N2 ) ) ) ).

% nat_less_le
tff(fact_236_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% less_add_eq_less
tff(fact_237_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_less_add2
tff(fact_238_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_less_add1
tff(fact_239_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_less_mono1
tff(fact_240_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

% not_add_less2
tff(fact_241_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

% not_add_less1
tff(fact_242_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_243_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% add_lessD1
tff(fact_244_nat__le__iff__add,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
    <=> ? [K3: nat] : N2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_245_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).

% trans_le_add2
tff(fact_246_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).

% trans_le_add1
tff(fact_247_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).

% add_le_mono1
tff(fact_248_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_le_mono
tff(fact_249_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_250_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) ) ).

% add_leD2
tff(fact_251_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% add_leD1
tff(fact_252_le__add2,axiom,
    ! [N2: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ).

% le_add2
tff(fact_253_le__add1,axiom,
    ! [N2: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M))) ).

% le_add1
tff(fact_254_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) ) ) ).

% add_leE
tff(fact_255_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_less_le_mono
tff(fact_256_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_less_le_mono
tff(fact_257_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_less_le_mono
tff(fact_258_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_less_le_mono
tff(fact_259_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).

% add_le_less_mono
tff(fact_260_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D))) ) ) ).

% add_le_less_mono
tff(fact_261_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).

% add_le_less_mono
tff(fact_262_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).

% add_le_less_mono
tff(fact_263_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_264_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_265_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_266_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_267_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_268_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_269_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_270_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_271_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,N2: nat,M: nat] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_272_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,N2: nat,M: nat] :
      ( ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_273_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,N2: nat,M: nat] :
      ( ! [N: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_274_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,N2: nat,M: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_275_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,N2: nat,M: nat] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,M)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_276_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N3))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_277_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_278_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_279_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_280_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_less
tff(fact_281_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N))),aa_nat_set_nat(F,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,N2))) ) ) ).

% lift_Suc_antimono_le
tff(fact_282_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,N2))) ) ) ).

% lift_Suc_antimono_le
tff(fact_283_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,N2))) ) ) ).

% lift_Suc_antimono_le
tff(fact_284_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,N2))) ) ) ).

% lift_Suc_antimono_le
tff(fact_285_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,N2))) ) ) ).

% lift_Suc_antimono_le
tff(fact_286_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_287_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_288_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_289_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_290_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,N2: nat,N3: nat] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,N3))) ) ) ).

% lift_Suc_mono_le
tff(fact_291_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N2))) ) ).

% le_imp_less_Suc
tff(fact_292_less__eq__Suc__le,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N2)),M)) ) ).

% less_eq_Suc_le
tff(fact_293_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% less_Suc_eq_le
tff(fact_294_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(suc,M)))
      <=> ( N2 = M ) ) ) ).

% le_less_Suc_eq
tff(fact_295_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% Suc_le_lessD
tff(fact_296_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N)))
                 => pp(aa_nat_bool(P,N)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_297_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J))
               => ( pp(aa_nat_bool(P,N))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_298_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% Suc_le_eq
tff(fact_299_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N2)) ) ).

% Suc_leI
tff(fact_300_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ? [K2: nat] : N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_301_less__iff__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
    <=> ? [K3: nat] : N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_302_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).

% less_add_Suc2
tff(fact_303_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).

% less_add_Suc1
tff(fact_304_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ~ ! [Q2: nat] : N2 != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ) ).

% less_natE
tff(fact_305_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M5)),aa_nat_nat(F,N))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_306_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N2))
     => ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
       => ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
          | pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).

% member_valid_both_member_options
tff(fact_307_discrete,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).

% discrete
tff(fact_308_discrete,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).

% discrete
tff(fact_309_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N2)),X)) ).

% buildup_nothing_in_leaf
tff(fact_310_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N2)),X)) ).

% buildup_nothing_in_min_max
tff(fact_311_add__mono1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real))) ) ).

% add_mono1
tff(fact_312_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat))) ) ).

% add_mono1
tff(fact_313_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat))) ) ).

% add_mono1
tff(fact_314_add__mono1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int))) ) ).

% add_mono1
tff(fact_315_less__add__one,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real))) ).

% less_add_one
tff(fact_316_less__add__one,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat))) ).

% less_add_one
tff(fact_317_less__add__one,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).

% less_add_one
tff(fact_318_less__add__one,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).

% less_add_one
tff(fact_319_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
      <=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_320_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
       => pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).

% valid_member_both_member_options
tff(fact_321_dual__order_Orefl,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).

% dual_order.refl
tff(fact_322_dual__order_Orefl,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% dual_order.refl
tff(fact_323_dual__order_Orefl,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% dual_order.refl
tff(fact_324_dual__order_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% dual_order.refl
tff(fact_325_dual__order_Orefl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% dual_order.refl
tff(fact_326_order__refl,axiom,
    ! [X: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(X),X)) ).

% order_refl
tff(fact_327_order__refl,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),X)) ).

% order_refl
tff(fact_328_order__refl,axiom,
    ! [X: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),X)) ).

% order_refl
tff(fact_329_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_330_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_331_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).

% min_Null_member
tff(fact_332_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
      <=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).

% member_correct
tff(fact_333_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_334_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_335_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_336_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( B != A ) ) ) ).

% nle_le
tff(fact_337_le__cases3,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2)) )
     => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z2)) )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z2))
           => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),Y)) )
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),Y))
             => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) )
           => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2))
               => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),X)) )
             => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),X))
                 => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_338_le__cases3,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
       => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z2)) )
     => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z2)) )
       => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z2))
           => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z2),Y)) )
         => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z2),Y))
             => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) )
           => ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z2))
               => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z2),X)) )
             => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z2),X))
                 => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_339_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z2)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z2))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),X)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),X))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_340_le__cases3,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z2)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z2))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),X)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),X))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_341_order__class_Oorder__eq__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_342_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_343_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_344_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_345_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_346_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_347_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_348_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_349_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_350_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_eq_le_trans
tff(fact_351_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( ( B = C )
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_352_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_353_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_354_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_355_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% ord_le_eq_trans
tff(fact_356_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_357_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_358_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_359_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_360_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_361_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),C)) ) ) ).

% order.trans
tff(fact_362_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),C)) ) ) ).

% order.trans
tff(fact_363_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),C)) ) ) ).

% order.trans
tff(fact_364_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C)) ) ) ).

% order.trans
tff(fact_365_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C)) ) ) ).

% order.trans
tff(fact_366_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Z2)) ) ) ).

% order_trans
tff(fact_367_order__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Z2)) ) ) ).

% order_trans
tff(fact_368_order__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Z2)) ) ) ).

% order_trans
tff(fact_369_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z2)) ) ) ).

% order_trans
tff(fact_370_order__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z2)) ) ) ).

% order_trans
tff(fact_371_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A3),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
     => ( ! [A3: rat,B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A3))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_372_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
     => ( ! [A3: num,B3: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
           => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
       => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_373_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_374_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ).

% linorder_wlog
tff(fact_375_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_376_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_377_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_378_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_379_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.eq_iff
tff(fact_380_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_381_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_382_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_383_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_384_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_385_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A)) ) ) ).

% dual_order.trans
tff(fact_386_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% dual_order.trans
tff(fact_387_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% dual_order.trans
tff(fact_388_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% dual_order.trans
tff(fact_389_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% dual_order.trans
tff(fact_390_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_391_antisym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_392_antisym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_393_antisym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_394_antisym,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => ( A = B ) ) ) ).

% antisym
tff(fact_395_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_396_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_397_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_398_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_399_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% Orderings.order_eq_iff
tff(fact_400_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_401_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_402_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_403_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_subst1
tff(fact_404_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_405_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_406_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_407_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_int_num(F,C))) ) ) ) ).

% order_subst1
tff(fact_408_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_409_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_subst1
tff(fact_410_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_411_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_412_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_413_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_414_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_415_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_416_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_417_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_418_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_419_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% order_subst2
tff(fact_420_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% order_eq_refl
tff(fact_421_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_eq_refl
tff(fact_422_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_eq_refl
tff(fact_423_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_424_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_eq_refl
tff(fact_425_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_linear
tff(fact_426_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_linear
tff(fact_427_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_linear
tff(fact_428_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_linear
tff(fact_429_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_430_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_431_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_432_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_433_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_434_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_435_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_num_nat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_436_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_num_int(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_437_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_438_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_nat_num(F,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_439_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_440_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_441_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_442_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_443_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_444_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_445_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_446_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_447_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_448_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,A)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_449_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_le_cases
tff(fact_450_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_le_cases
tff(fact_451_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_le_cases
tff(fact_452_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_le_cases
tff(fact_453_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_454_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_455_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_456_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_457_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_458_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X)) ).

% lt_ex
tff(fact_459_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X)) ).

% lt_ex
tff(fact_460_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y3),X)) ).

% lt_ex
tff(fact_461_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X_1)) ).

% gt_ex
tff(fact_462_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X_1)) ).

% gt_ex
tff(fact_463_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_1)) ).

% gt_ex
tff(fact_464_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_1)) ).

% gt_ex
tff(fact_465_dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Z: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).

% dense
tff(fact_466_dense,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ? [Z: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).

% dense
tff(fact_467_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_468_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_469_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_470_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_471_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_472_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order.asym
tff(fact_473_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order.asym
tff(fact_474_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order.asym
tff(fact_475_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order.asym
tff(fact_476_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order.asym
tff(fact_477_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_478_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_479_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_480_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_481_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_eq_less_trans
tff(fact_482_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( B = C )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_483_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( B = C )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_484_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( ( B = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_485_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( ( B = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_486_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( ( B = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% ord_less_eq_trans
tff(fact_487_less__induct,axiom,
    ! [P: fun_nat_bool,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y4),X4))
             => pp(aa_nat_bool(P,Y4)) )
         => pp(aa_nat_bool(P,X4)) )
     => pp(aa_nat_bool(P,A)) ) ).

% less_induct
tff(fact_488_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_489_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_490_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_491_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_492_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_493_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ( X != Y )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_cases
tff(fact_494_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ( X != Y )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_cases
tff(fact_495_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( ( X != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_cases
tff(fact_496_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_497_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_498_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% dual_order.asym
tff(fact_499_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% dual_order.asym
tff(fact_500_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% dual_order.asym
tff(fact_501_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% dual_order.asym
tff(fact_502_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% dual_order.asym
tff(fact_503_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% dual_order.irrefl
tff(fact_504_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% dual_order.irrefl
tff(fact_505_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% dual_order.irrefl
tff(fact_506_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% dual_order.irrefl
tff(fact_507_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% dual_order.irrefl
tff(fact_508_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [N4: nat] :
          ( pp(aa_nat_bool(P,N4))
          & ! [M6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N4))
             => ~ pp(aa_nat_bool(P,M6)) ) ) ) ).

% exists_least_iff
tff(fact_509_linorder__less__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
     => ( ! [A3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),A3))
       => ( ! [A3: real,B3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),A3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_510_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
     => ( ! [A3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),A3))
       => ( ! [A3: rat,B3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(P,B3),A3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A3),B3)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_511_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A3),A3))
       => ( ! [A3: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_512_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_513_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A),B)) ) ) ) ).

% linorder_less_wlog
tff(fact_514_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans
tff(fact_515_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans
tff(fact_516_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans
tff(fact_517_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans
tff(fact_518_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans
tff(fact_519_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_520_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_521_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_522_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_523_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_524_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_525_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_526_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_527_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_528_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans
tff(fact_529_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_530_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_531_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_532_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_533_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_534_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_535_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_536_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_537_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_538_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_539_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_540_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_541_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_542_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neqE
tff(fact_543_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_544_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neqE
tff(fact_545_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_546_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE
tff(fact_547_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_asym
tff(fact_548_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_asym
tff(fact_549_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_asym
tff(fact_550_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_asym
tff(fact_551_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_asym
tff(fact_552_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_553_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_554_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_555_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_556_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_557_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% order_less_asym'
tff(fact_558_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% order_less_asym'
tff(fact_559_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A)) ) ).

% order_less_asym'
tff(fact_560_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A)) ) ).

% order_less_asym'
tff(fact_561_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% order_less_asym'
tff(fact_562_order__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2)) ) ) ).

% order_less_trans
tff(fact_563_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2)) ) ) ).

% order_less_trans
tff(fact_564_order__less__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2)) ) ) ).

% order_less_trans
tff(fact_565_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) ) ) ).

% order_less_trans
tff(fact_566_order__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2)) ) ) ).

% order_less_trans
tff(fact_567_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_568_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_569_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_real_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_570_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_real_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_571_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_real_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_572_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_573_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_574_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_575_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_576_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_577_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_578_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_579_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_580_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_581_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_582_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_583_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_584_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_585_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_586_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_587_order__less__irrefl,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X)) ).

% order_less_irrefl
tff(fact_588_order__less__irrefl,axiom,
    ! [X: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),X)) ).

% order_less_irrefl
tff(fact_589_order__less__irrefl,axiom,
    ! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).

% order_less_irrefl
tff(fact_590_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_591_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_592_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_593_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_594_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_595_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_596_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_597_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_598_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_599_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_600_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_601_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_less_subst1
tff(fact_602_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_603_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_604_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_real_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_605_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_606_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_607_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_608_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_609_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_610_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_611_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_less_subst2
tff(fact_612_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_not_sym
tff(fact_613_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_not_sym
tff(fact_614_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_not_sym
tff(fact_615_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_not_sym
tff(fact_616_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_not_sym
tff(fact_617_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_618_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: bool] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_619_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_620_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_621_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_622_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      | ( X = Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_less_linear
tff(fact_623_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      | ( X = Y )
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_less_linear
tff(fact_624_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      | ( X = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_less_linear
tff(fact_625_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_626_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_627_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_628_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_629_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_630_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_631_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_632_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_633_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_634_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_635_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_636_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_637_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_638_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_639_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_640_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_641_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_642_bounded__Max__nat,axiom,
    ! [P: fun_nat_bool,X: nat,M7: nat] :
      ( pp(aa_nat_bool(P,X))
     => ( ! [X4: nat] :
            ( pp(aa_nat_bool(P,X4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M7)) )
       => ~ ! [M5: nat] :
              ( pp(aa_nat_bool(P,M5))
             => ~ ! [X2: nat] :
                    ( pp(aa_nat_bool(P,X2))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_643_leD,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% leD
tff(fact_644_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y)) ) ).

% leD
tff(fact_645_leD,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% leD
tff(fact_646_leD,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% leD
tff(fact_647_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_648_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_649_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% leI
tff(fact_650_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% leI
tff(fact_651_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% leI
tff(fact_652_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% leI
tff(fact_653_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% leI
tff(fact_654_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_655_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_656_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_657_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_658_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_659_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        | ( A = B ) ) ) ).

% nless_le
tff(fact_660_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_661_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_662_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_663_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_664_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_665_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_666_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_667_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( ~ pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_668_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_669_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_670_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_671_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_672_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X4)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ).

% dense_ge
tff(fact_673_dense__ge,axiom,
    ! [Z2: rat,Y: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X4)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2)) ) ).

% dense_ge
tff(fact_674_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z2)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ).

% dense_le
tff(fact_675_dense__le,axiom,
    ! [Y: rat,Z2: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Z2)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2)) ) ).

% dense_le
tff(fact_676_less__le__not__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ).

% less_le_not_le
tff(fact_677_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ) ).

% less_le_not_le
tff(fact_678_less__le__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_679_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% less_le_not_le
tff(fact_680_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_681_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_682_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% not_le_imp_less
tff(fact_683_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).

% not_le_imp_less
tff(fact_684_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% not_le_imp_less
tff(fact_685_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_686_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_687_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_688_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_689_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_690_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_691_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_692_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_693_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_694_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_695_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_696_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_697_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_698_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_699_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans1
tff(fact_700_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).

% order.strict_trans1
tff(fact_701_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_702_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans1
tff(fact_703_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans1
tff(fact_704_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans1
tff(fact_705_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C)) ) ) ).

% order.strict_trans2
tff(fact_706_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),C))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),C)) ) ) ).

% order.strict_trans2
tff(fact_707_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_708_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),C)) ) ) ).

% order.strict_trans2
tff(fact_709_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),C)) ) ) ).

% order.strict_trans2
tff(fact_710_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),C)) ) ) ).

% order.strict_trans2
tff(fact_711_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_712_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ) ).

% order.strict_iff_not
tff(fact_713_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_714_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_715_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_716_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% order.strict_iff_not
tff(fact_717_dense__ge__bounded,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),X))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),W)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ) ).

% dense_ge_bounded
tff(fact_718_dense__ge__bounded,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),X))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),W)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2)) ) ) ).

% dense_ge_bounded
tff(fact_719_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( ! [W: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),W))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Y))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z2)) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2)) ) ) ).

% dense_le_bounded
tff(fact_720_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( ! [W: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),W))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Y))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z2)) ) )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2)) ) ) ).

% dense_le_bounded
tff(fact_721_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_722_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_723_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_724_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_725_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_726_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_727_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_728_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_729_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_730_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_731_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_732_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_733_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_734_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_735_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_736_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_737_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_738_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans1
tff(fact_739_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_740_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_741_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_742_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_743_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_744_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% dual_order.strict_trans2
tff(fact_745_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_746_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_747_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_748_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_749_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_750_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% dual_order.strict_iff_not
tff(fact_751_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% order.strict_implies_order
tff(fact_752_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A),B))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% order.strict_implies_order
tff(fact_753_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% order.strict_implies_order
tff(fact_754_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% order.strict_implies_order
tff(fact_755_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% order.strict_implies_order
tff(fact_756_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% order.strict_implies_order
tff(fact_757_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_758_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(B),A))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_759_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_760_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_761_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_762_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% dual_order.strict_implies_order
tff(fact_763_order__le__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_764_order__le__less,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_765_order__le__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_766_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_767_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_768_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_769_order__less__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_770_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_771_order__less__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_772_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_773_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_774_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_775_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_not_le
tff(fact_776_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_not_le
tff(fact_777_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_not_le
tff(fact_778_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_779_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_780_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ).

% linorder_not_less
tff(fact_781_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X)) ) ).

% linorder_not_less
tff(fact_782_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_not_less
tff(fact_783_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_784_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_785_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% order_less_imp_le
tff(fact_786_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% order_less_imp_le
tff(fact_787_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% order_less_imp_le
tff(fact_788_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_less_imp_le
tff(fact_789_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_790_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_791_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ( A != B )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_792_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
     => ( ( A != B )
       => pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).

% order_le_neq_trans
tff(fact_793_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( ( A != B )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_794_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( ( A != B )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_795_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( ( A != B )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_796_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( ( A != B )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_le_neq_trans
tff(fact_797_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_798_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => pp(aa_set_nat_bool(ord_less_set_nat(A),B)) ) ) ).

% order_neq_le_trans
tff(fact_799_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_800_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_801_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_802_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% order_neq_le_trans
tff(fact_803_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_804_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(Y),Z2))
       => pp(aa_set_nat_bool(ord_less_set_nat(X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_805_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_806_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_807_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_808_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2)) ) ) ).

% order_le_less_trans
tff(fact_809_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_810_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),Z2))
       => pp(aa_set_nat_bool(ord_less_set_nat(X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_811_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),Z2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_812_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_813_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_814_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2)) ) ) ).

% order_less_le_trans
tff(fact_815_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_816_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_817_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_818_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_nat_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_819_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_int_real(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_int_real(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_820_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_real_rat(F,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_real_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_821_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_822_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_823_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(F,B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_nat_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_824_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_int_rat(F,B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_int_rat(F,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_825_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_826_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_827_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_rat_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_828_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_rat_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_829_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_rat_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_830_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_831_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_832_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_833_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_834_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(F,A)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_835_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_rat_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_836_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_837_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_rat_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_838_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_rat_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_839_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_rat_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_840_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_num_real(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_841_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_num_rat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_842_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y3))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),aa_num_num(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_843_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_num_nat(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_844_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,B)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Y3))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y3))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_num_int(F,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_845_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_846_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_847_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_848_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_849_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,B)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_850_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,B)),C))
       => ( ! [X4: real,Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_851_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(F,B)),C))
       => ( ! [X4: rat,Y3: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_852_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(F,B)),C))
       => ( ! [X4: num,Y3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_853_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,B)),C))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_854_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,B)),C))
       => ( ! [X4: int,Y3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Y3))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,A)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_855_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_856_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
      | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_857_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_858_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_859_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_860_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_861_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_862_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_863_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_864_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_865_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_866_buildup__gives__valid,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_buildup(N2)),N2)) ) ).

% buildup_gives_valid
tff(fact_867_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
    <=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq
tff(fact_868_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),D))
     => pp(aa_nat_bool(vEBT_VEBT_valid(T),D)) ) ).

% valid_eq1
tff(fact_869_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
     => pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).

% valid_eq2
tff(fact_870_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% deg_not_0
tff(fact_871_nat__descend__induct,axiom,
    ! [N2: nat,P: fun_nat_bool,M: nat] :
      ( ! [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K2))
         => pp(aa_nat_bool(P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N2))
           => ( ! [I4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),I4))
                 => pp(aa_nat_bool(P,I4)) )
             => pp(aa_nat_bool(P,K2)) ) )
       => pp(aa_nat_bool(P,M)) ) ) ).

% nat_descend_induct
tff(fact_872_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).

% less_numeral_extra(4)
tff(fact_873_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_874_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_875_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_876_le__numeral__extra_I4_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).

% le_numeral_extra(4)
tff(fact_877_le__numeral__extra_I4_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).

% le_numeral_extra(4)
tff(fact_878_le__numeral__extra_I4_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).

% le_numeral_extra(4)
tff(fact_879_le__numeral__extra_I4_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).

% le_numeral_extra(4)
tff(fact_880_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X2)) ) ).

% minf(8)
tff(fact_881_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X2)) ) ).

% minf(8)
tff(fact_882_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X2)) ) ).

% minf(8)
tff(fact_883_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X2)) ) ).

% minf(8)
tff(fact_884_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2)) ) ).

% minf(8)
tff(fact_885_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),T)) ) ).

% minf(6)
tff(fact_886_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),T)) ) ).

% minf(6)
tff(fact_887_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),T)) ) ).

% minf(6)
tff(fact_888_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),T)) ) ).

% minf(6)
tff(fact_889_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T)) ) ).

% minf(6)
tff(fact_890_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X2)) ) ).

% pinf(8)
tff(fact_891_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X2)) ) ).

% pinf(8)
tff(fact_892_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X2)) ) ).

% pinf(8)
tff(fact_893_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X2)) ) ).

% pinf(8)
tff(fact_894_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2)) ) ).

% pinf(8)
tff(fact_895_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_tree_deg_neq_0
tff(fact_896_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).

% valid_0_not
tff(fact_897_le__zero__eq,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),zero_zero_nat))
    <=> ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_898_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
    <=> ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_899_add__0,axiom,
    ! [A: literal] : plus_plus_literal(zero_zero_literal,A) = A ).

% add_0
tff(fact_900_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_901_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_902_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_903_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_904_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_905_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_906_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_907_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_908_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_909_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_910_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_911_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_912_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_913_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_914_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_915_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_916_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_917_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_918_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_919_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_920_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_921_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_922_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_923_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_924_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_925_add_Oright__neutral,axiom,
    ! [A: literal] : plus_plus_literal(A,zero_zero_literal) = A ).

% add.right_neutral
tff(fact_926_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_927_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_928_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_929_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_930_less__nat__zero__code,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_931_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% neq0_conv
tff(fact_932_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_933_le0,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N2)) ).

% le0
tff(fact_934_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).

% bot_nat_0.extremum
tff(fact_935_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_936_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_937_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_938_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_939_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_940_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_941_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_942_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_943_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel2
tff(fact_944_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel2
tff(fact_945_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel2
tff(fact_946_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel2
tff(fact_947_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).

% le_add_same_cancel1
tff(fact_948_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).

% le_add_same_cancel1
tff(fact_949_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).

% le_add_same_cancel1
tff(fact_950_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).

% le_add_same_cancel1
tff(fact_951_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel2
tff(fact_952_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_953_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_954_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_955_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% add_le_same_cancel1
tff(fact_956_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_957_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_958_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_959_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_960_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_961_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_962_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_963_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_964_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_965_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel2
tff(fact_966_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel2
tff(fact_967_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel2
tff(fact_968_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel2
tff(fact_969_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).

% less_add_same_cancel1
tff(fact_970_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).

% less_add_same_cancel1
tff(fact_971_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).

% less_add_same_cancel1
tff(fact_972_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).

% less_add_same_cancel1
tff(fact_973_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel2
tff(fact_974_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_975_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_976_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_977_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% add_less_same_cancel1
tff(fact_978_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_979_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_980_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_981_zero__less__Suc,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N2))) ).

% zero_less_Suc
tff(fact_982_less__Suc0,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N2 = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_983_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% add_gr_0
tff(fact_984_less__one,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),one_one_nat))
    <=> ( N2 = zero_zero_nat ) ) ).

% less_one
tff(fact_985_zero__reorient,axiom,
    ! [X: literal] :
      ( ( zero_zero_literal = X )
    <=> ( X = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_986_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_987_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_988_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_989_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_990_le__numeral__extra_I3_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).

% le_numeral_extra(3)
tff(fact_991_le__numeral__extra_I3_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).

% le_numeral_extra(3)
tff(fact_992_le__numeral__extra_I3_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).

% le_numeral_extra(3)
tff(fact_993_le__numeral__extra_I3_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% le_numeral_extra(3)
tff(fact_994_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).

% less_numeral_extra(3)
tff(fact_995_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_996_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_997_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_998_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_999_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
    <=> ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1000_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1001_not__less__zero,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),zero_zero_nat)) ).

% not_less_zero
tff(fact_1002_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% gr_zeroI
tff(fact_1003_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_1004_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_1005_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_1006_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_1007_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_1008_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_1009_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_1010_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_1011_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1012_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_1013_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_1014_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1015_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1016_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1017_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1018_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1019_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] : N2 = aa_nat_nat(suc,M5) ) ).

% not0_implies_Suc
tff(fact_1020_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1021_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1022_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1023_zero__induct,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N)))
           => pp(aa_nat_bool(P,N)) )
       => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).

% zero_induct
tff(fact_1024_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N2: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),zero_zero_nat))
     => ( ! [Y3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y3)))
       => ( ! [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X4),Y3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N2)) ) ) ) ).

% diff_induct
tff(fact_1025_nat__induct,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(P,N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N))) )
       => pp(aa_nat_bool(P,N2)) ) ) ).

% nat_induct
tff(fact_1026_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1027_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1028_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1029_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1030_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_1031_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( ~ pp(aa_nat_bool(P,N))
             => ? [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
                  & ~ pp(aa_nat_bool(P,M2)) ) ) )
       => pp(aa_nat_bool(P,N2)) ) ) ).

% infinite_descent0
tff(fact_1032_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1033_less__zeroE,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),zero_zero_nat)) ).

% less_zeroE
tff(fact_1034_not__less0,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),zero_zero_nat)) ).

% not_less0
tff(fact_1035_not__gr0,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
    <=> ( N2 = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1036_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% gr0I
tff(fact_1037_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_1038_le__0__eq,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),zero_zero_nat))
    <=> ( N2 = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1039_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1040_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1041_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N2)) ).

% less_eq_nat.simps(1)
tff(fact_1042_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1043_plus__nat_Oadd__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N2) = N2 ).

% plus_nat.add_0
tff(fact_1044_less__numeral__extra_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% less_numeral_extra(1)
tff(fact_1045_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_1046_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_1047_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_1048_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1049_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1050_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1051_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1052_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1053_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_rat_rat(plus_plus_rat(X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1054_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1055_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1056_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_nonpos
tff(fact_1057_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1058_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_1059_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_1060_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1061_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1062_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1063_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_nonneg
tff(fact_1064_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing2
tff(fact_1065_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_increasing2
tff(fact_1066_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing2
tff(fact_1067_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing2
tff(fact_1068_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1069_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1070_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1071_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing2
tff(fact_1072_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_increasing
tff(fact_1073_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_increasing
tff(fact_1074_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_increasing
tff(fact_1075_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_increasing
tff(fact_1076_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1077_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1078_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1079_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).

% add_decreasing
tff(fact_1080_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% zero_less_one_class.zero_le_one
tff(fact_1081_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1082_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_1083_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_1084_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1085_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1086_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1087_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1088_not__one__le__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).

% not_one_le_zero
tff(fact_1089_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_1090_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_1091_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_1092_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% pos_add_strict
tff(fact_1093_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% pos_add_strict
tff(fact_1094_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% pos_add_strict
tff(fact_1095_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% pos_add_strict
tff(fact_1096_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1097_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_pos
tff(fact_1098_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_pos_pos
tff(fact_1099_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_pos
tff(fact_1100_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_pos
tff(fact_1101_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_neg
tff(fact_1102_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_1103_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_1104_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_1105_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).

% add_less_zeroD
tff(fact_1106_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(X),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_1107_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_1108_zero__less__one,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).

% zero_less_one
tff(fact_1109_zero__less__one,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_1110_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_1111_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_1112_not__one__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).

% not_one_less_zero
tff(fact_1113_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_1114_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_1115_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_1116_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N2)))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N2)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1117_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ? [M5: nat] : N2 = aa_nat_nat(suc,M5) ) ).

% gr0_implies_Suc
tff(fact_1118_All__less__Suc2,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N2)))
         => pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N2))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% All_less_Suc2
tff(fact_1119_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
    <=> ? [M6: nat] : N2 = aa_nat_nat(suc,M6) ) ).

% gr0_conv_Suc
tff(fact_1120_Ex__less__Suc2,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(suc,N2)))
          & pp(aa_nat_bool(P,I3)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),N2))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I3))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1121_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1122_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1123_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( pp(aa_nat_bool(P,N2))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N2))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_1124_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K2))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1125_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1126_pinf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q3,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
           => ( ( pp(aa_real_bool(P,X2))
                & pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                & pp(aa_real_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1127_pinf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q3,X4)) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
           => ( ( pp(aa_rat_bool(P,X2))
                & pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P2,X2))
                & pp(aa_rat_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1128_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q3,X4)) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
           => ( ( pp(aa_num_bool(P,X2))
                & pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P2,X2))
                & pp(aa_num_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1129_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q3,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
           => ( ( pp(aa_nat_bool(P,X2))
                & pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                & pp(aa_nat_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1130_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q3,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                & pp(aa_int_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(1)
tff(fact_1131_pinf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q3,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
           => ( ( pp(aa_real_bool(P,X2))
                | pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                | pp(aa_real_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1132_pinf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q3,X4)) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
           => ( ( pp(aa_rat_bool(P,X2))
                | pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P2,X2))
                | pp(aa_rat_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1133_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q3,X4)) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
           => ( ( pp(aa_num_bool(P,X2))
                | pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P2,X2))
                | pp(aa_num_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1134_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q3,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
           => ( ( pp(aa_nat_bool(P,X2))
                | pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                | pp(aa_nat_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1135_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q3,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                | pp(aa_int_bool(Q3,X2)) ) ) ) ) ) ).

% pinf(2)
tff(fact_1136_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1137_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1138_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1139_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1140_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_1141_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1142_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1143_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1144_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1145_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_1146_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T)) ) ).

% pinf(5)
tff(fact_1147_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),T)) ) ).

% pinf(5)
tff(fact_1148_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),T)) ) ).

% pinf(5)
tff(fact_1149_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),T)) ) ).

% pinf(5)
tff(fact_1150_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T)) ) ).

% pinf(5)
tff(fact_1151_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X2)) ) ).

% pinf(7)
tff(fact_1152_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X2)) ) ).

% pinf(7)
tff(fact_1153_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X2)) ) ).

% pinf(7)
tff(fact_1154_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X2)) ) ).

% pinf(7)
tff(fact_1155_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2)) ) ).

% pinf(7)
tff(fact_1156_minf_I1_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q3,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
           => ( ( pp(aa_real_bool(P,X2))
                & pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                & pp(aa_real_bool(Q3,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1157_minf_I1_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q3,X4)) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
           => ( ( pp(aa_rat_bool(P,X2))
                & pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P2,X2))
                & pp(aa_rat_bool(Q3,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1158_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q3,X4)) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
           => ( ( pp(aa_num_bool(P,X2))
                & pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P2,X2))
                & pp(aa_num_bool(Q3,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1159_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q3,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
           => ( ( pp(aa_nat_bool(P,X2))
                & pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                & pp(aa_nat_bool(Q3,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1160_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q3,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                & pp(aa_int_bool(Q3,X2)) ) ) ) ) ) ).

% minf(1)
tff(fact_1161_minf_I2_J,axiom,
    ! [P: fun_real_bool,P2: fun_real_bool,Q: fun_real_bool,Q3: fun_real_bool] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
         => ( pp(aa_real_bool(P,X4))
          <=> pp(aa_real_bool(P2,X4)) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
           => ( pp(aa_real_bool(Q,X4))
            <=> pp(aa_real_bool(Q3,X4)) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
           => ( ( pp(aa_real_bool(P,X2))
                | pp(aa_real_bool(Q,X2)) )
            <=> ( pp(aa_real_bool(P2,X2))
                | pp(aa_real_bool(Q3,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1162_minf_I2_J,axiom,
    ! [P: fun_rat_bool,P2: fun_rat_bool,Q: fun_rat_bool,Q3: fun_rat_bool] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
         => ( pp(aa_rat_bool(P,X4))
          <=> pp(aa_rat_bool(P2,X4)) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
           => ( pp(aa_rat_bool(Q,X4))
            <=> pp(aa_rat_bool(Q3,X4)) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
           => ( ( pp(aa_rat_bool(P,X2))
                | pp(aa_rat_bool(Q,X2)) )
            <=> ( pp(aa_rat_bool(P2,X2))
                | pp(aa_rat_bool(Q3,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1163_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q3: fun_num_bool] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
         => ( pp(aa_num_bool(P,X4))
          <=> pp(aa_num_bool(P2,X4)) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
           => ( pp(aa_num_bool(Q,X4))
            <=> pp(aa_num_bool(Q3,X4)) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
           => ( ( pp(aa_num_bool(P,X2))
                | pp(aa_num_bool(Q,X2)) )
            <=> ( pp(aa_num_bool(P2,X2))
                | pp(aa_num_bool(Q3,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1164_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q3: fun_nat_bool] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
         => ( pp(aa_nat_bool(P,X4))
          <=> pp(aa_nat_bool(P2,X4)) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
           => ( pp(aa_nat_bool(Q,X4))
            <=> pp(aa_nat_bool(Q3,X4)) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
           => ( ( pp(aa_nat_bool(P,X2))
                | pp(aa_nat_bool(Q,X2)) )
            <=> ( pp(aa_nat_bool(P2,X2))
                | pp(aa_nat_bool(Q3,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1165_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q3: fun_int_bool] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
         => ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P2,X4)) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(Q,X4))
            <=> pp(aa_int_bool(Q3,X4)) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
            <=> ( pp(aa_int_bool(P2,X2))
                | pp(aa_int_bool(Q3,X2)) ) ) ) ) ) ).

% minf(2)
tff(fact_1166_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1167_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1168_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1169_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1170_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_1171_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1172_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1173_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1174_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1175_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_1176_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),T)) ) ).

% minf(5)
tff(fact_1177_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),T)) ) ).

% minf(5)
tff(fact_1178_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),T)) ) ).

% minf(5)
tff(fact_1179_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),T)) ) ).

% minf(5)
tff(fact_1180_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T)) ) ).

% minf(5)
tff(fact_1181_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X2)) ) ).

% minf(7)
tff(fact_1182_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X2)) ) ).

% minf(7)
tff(fact_1183_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X2)) ) ).

% minf(7)
tff(fact_1184_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X2)) ) ).

% minf(7)
tff(fact_1185_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2)) ) ).

% minf(7)
tff(fact_1186_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_1187_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_1188_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_1189_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1190_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1191_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1192_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing2
tff(fact_1193_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1194_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C))) ) ) ).

% add_strict_increasing
tff(fact_1195_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1196_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).

% add_strict_increasing
tff(fact_1197_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1198_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1199_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1200_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_pos_nonneg
tff(fact_1201_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_nonpos_neg
tff(fact_1202_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_1203_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_1204_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_1205_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1206_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1207_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1208_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% add_nonneg_pos
tff(fact_1209_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).

% add_neg_nonpos
tff(fact_1210_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_1211_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_1212_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_1213_zero__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ).

% zero_less_two
tff(fact_1214_zero__less__two,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ).

% zero_less_two
tff(fact_1215_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% zero_less_two
tff(fact_1216_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_1217_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( pp(aa_nat_bool(P,N2))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N2))
            & ! [I4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),K2))
               => ~ pp(aa_nat_bool(P,I4)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_1218_nat__induct__non__zero,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
             => ( pp(aa_nat_bool(P,N))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N))) ) )
         => pp(aa_nat_bool(P,N2)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1219_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),T)) ) ).

% pinf(6)
tff(fact_1220_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),T)) ) ).

% pinf(6)
tff(fact_1221_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),T)) ) ).

% pinf(6)
tff(fact_1222_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),T)) ) ).

% pinf(6)
tff(fact_1223_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T)) ) ).

% pinf(6)
tff(fact_1224_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_1225_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_1226_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1227_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_1228_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_epsilon
tff(fact_1229_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(Y),E))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_epsilon
tff(fact_1230_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),zero_zero_nat))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).

% Euclid_induct
tff(fact_1231_exists__least__lemma,axiom,
    ! [P: fun_nat_bool] :
      ( ~ pp(aa_nat_bool(P,zero_zero_nat))
     => ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
       => ? [N: nat] :
            ( ~ pp(aa_nat_bool(P,N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,N))) ) ) ) ).

% exists_least_lemma
tff(fact_1232_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_1233_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] : X != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_1234_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_1235_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1236_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_1237_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_1238_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_1239_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_1240_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_1241_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_1242_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
       => ? [E: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D1))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1243_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D2))
       => ? [E: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D1))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E),D2)) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1244_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [Z4: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z4),Y)) ) ) ) ).

% pred_member
tff(fact_1245_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [Z4: nat] :
            ( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z4)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z4)) ) ) ) ).

% succ_member
tff(fact_1246_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1247_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1248_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1249_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1250_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).

% verit_comp_simplify1(2)
tff(fact_1251_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      | ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% verit_la_disequality
tff(fact_1252_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
      | ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% verit_la_disequality
tff(fact_1253_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
      | ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% verit_la_disequality
tff(fact_1254_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      | ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% verit_la_disequality
tff(fact_1255_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1256_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1257_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1258_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1259_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).

% verit_comp_simplify1(1)
tff(fact_1260_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),X2)) ).

% linordered_field_no_lb
tff(fact_1261_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y3: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y3),X2)) ).

% linordered_field_no_lb
tff(fact_1262_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),X_1)) ).

% linordered_field_no_ub
tff(fact_1263_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),X_1)) ).

% linordered_field_no_ub
tff(fact_1264_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),A4))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1265_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B4),A4))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1266_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B4),A4))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1267_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B4),A4))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1268_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),A4))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A4),B4)) ) ).

% verit_comp_simplify1(3)
tff(fact_1269_buildup__gives__empty,axiom,
    ! [N2: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N2)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1270_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).

% set_vebt_finite
tff(fact_1271_triangle__Suc,axiom,
    ! [N2: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N2)),aa_nat_nat(suc,N2)) ).

% triangle_Suc
tff(fact_1272_complete__interval,axiom,
    ! [A: real,B: real,P: fun_real_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(P,A))
       => ( ~ pp(aa_real_bool(P,B))
         => ? [C3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C3),B))
              & ! [X2: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X2))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),C3)) )
                 => pp(aa_real_bool(P,X2)) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),D3)) )
                     => pp(aa_real_bool(P,X4)) )
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1273_complete__interval,axiom,
    ! [A: nat,B: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(P,A))
       => ( ~ pp(aa_nat_bool(P,B))
         => ? [C3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C3))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C3),B))
              & ! [X2: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X2))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),C3)) )
                 => pp(aa_nat_bool(P,X2)) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),D3)) )
                     => pp(aa_nat_bool(P,X4)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1274_complete__interval,axiom,
    ! [A: int,B: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(P,A))
       => ( ~ pp(aa_int_bool(P,B))
         => ? [C3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),C3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C3),B))
              & ! [X2: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X2))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),C3)) )
                 => pp(aa_int_bool(P,X2)) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),D3)) )
                     => pp(aa_int_bool(P,X4)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D3),C3)) ) ) ) ) ) ).

% complete_interval
tff(fact_1275_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_1276_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_1277_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_1278_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_1279_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N2))))
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = vEBT_Node(Info,aa_nat_nat(suc,aa_nat_nat(suc,N2)),TreeList,S2) ) ).

% deg_SUcn_Node
tff(fact_1280_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
    <=> ? [A5: bool,B5: bool] : T = vEBT_Leaf(A5,B5) ) ).

% deg1Leaf
tff(fact_1281_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
     => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ).

% deg_1_Leaf
tff(fact_1282_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( N2 = one_one_nat )
       => ? [A3: bool,B3: bool] : T = vEBT_Leaf(A3,B3) ) ) ).

% deg_1_Leafy
tff(fact_1283_Leaf__0__not,axiom,
    ! [A: bool,B: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),zero_zero_nat)) ).

% Leaf_0_not
tff(fact_1284_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% frac_eq
tff(fact_1285_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ) ).

% frac_eq
tff(fact_1286_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N2))
     => ( Deg = N2 ) ) ).

% deg_deg_n
tff(fact_1287_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A),X_1))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X2)) ) ) ) ).

% succ_none_empty
tff(fact_1288_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A),X_1))
     => ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),Xs))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),A)) ) ) ) ).

% pred_none_empty
tff(fact_1289_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_1290_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_1291_obtain__set__succ,axiom,
    ! [X: nat,Z2: nat,A2: set_nat,B2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(A2),Z2))
       => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
         => ( ( A2 = B2 )
           => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A2),X),X_1)) ) ) ) ) ).

% obtain_set_succ
tff(fact_1292_obtain__set__pred,axiom,
    ! [Z2: nat,X: nat,A2: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(A2),Z2))
       => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
         => ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A2),X),X_1)) ) ) ) ).

% obtain_set_pred
tff(fact_1293_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1294_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_1295_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: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_1296_ex__min__if__finite,axiom,
    ! [S3: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
            & ~ ? [Xa: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),S3))
                  & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1297_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)
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1298_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)
                  & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1299_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)
                  & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1300_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),S3))
            & ~ ? [Xa: nat] :
                  ( pp(aa_set_nat_bool(member_nat(Xa),S3))
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1301_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),S3))
            & ~ ? [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),S3))
                  & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X4)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1302_infinite__growing,axiom,
    ! [X5: set_Extended_enat] :
      ( ( X5 != bot_bo7653980558646680370d_enat )
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),X5))
           => ? [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),X5))
                & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X4),Xa)) ) )
       => ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X5)) ) ) ).

% infinite_growing
tff(fact_1303_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( member_real(X4,X5)
           => ? [Xa: real] :
                ( member_real(Xa,X5)
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Xa)) ) )
       => ~ finite_finite_real(X5) ) ) ).

% infinite_growing
tff(fact_1304_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( member_rat(X4,X5)
           => ? [Xa: rat] :
                ( member_rat(Xa,X5)
                & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Xa)) ) )
       => ~ finite_finite_rat(X5) ) ) ).

% infinite_growing
tff(fact_1305_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( member_num(X4,X5)
           => ? [Xa: num] :
                ( member_num(Xa,X5)
                & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Xa)) ) )
       => ~ finite_finite_num(X5) ) ) ).

% infinite_growing
tff(fact_1306_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),X5))
           => ? [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),X5))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa)) ) )
       => ~ pp(aa_set_nat_bool(finite_finite_nat,X5)) ) ) ).

% infinite_growing
tff(fact_1307_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),X5))
           => ? [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),X5))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Xa)) ) )
       => ~ pp(aa_set_int_bool(finite_finite_int,X5)) ) ) ).

% infinite_growing
tff(fact_1308_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat))
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
tff(fact_1309_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1310_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1311_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),bot_bot_set_nat))
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1312_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1313_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A),bot_bo7653980558646680370d_enat))
    <=> ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
tff(fact_1314_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1315_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(A),bot_bot_set_int))
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1316_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),bot_bot_set_nat))
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1317_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),bot_bot_nat))
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1318_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(bot_bo7653980558646680370d_enat),A)) ).

% bot.extremum
tff(fact_1319_bot_Oextremum,axiom,
    ! [A: set_real] : ord_less_eq_set_real(bot_bot_set_real,A) ).

% bot.extremum
tff(fact_1320_bot_Oextremum,axiom,
    ! [A: set_int] : pp(aa_set_int_bool(ord_less_eq_set_int(bot_bot_set_int),A)) ).

% bot.extremum
tff(fact_1321_bot_Oextremum,axiom,
    ! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(bot_bot_set_nat),A)) ).

% bot.extremum
tff(fact_1322_bot_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A)) ).

% bot.extremum
tff(fact_1323_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
    <=> pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(bot_bo7653980558646680370d_enat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1324_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> ord_less_set_real(bot_bot_set_real,A) ) ).

% bot.not_eq_extremum
tff(fact_1325_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_set_nat(bot_bot_set_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1326_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_set_int(bot_bot_set_int),A)) ) ).

% bot.not_eq_extremum
tff(fact_1327_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A)) ) ).

% bot.not_eq_extremum
tff(fact_1328_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] : ~ pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A),bot_bo7653980558646680370d_enat)) ).

% bot.extremum_strict
tff(fact_1329_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ ord_less_set_real(A,bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_1330_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(A),bot_bot_set_nat)) ).

% bot.extremum_strict
tff(fact_1331_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A),bot_bot_set_int)) ).

% bot.extremum_strict
tff(fact_1332_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_1333_finite__nat__set__iff__bounded,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
    <=> ? [M6: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),M6)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1334_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),N2)) )
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% bounded_nat_set_is_finite
tff(fact_1335_finite__nat__set__iff__bounded__le,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
    <=> ? [M6: nat] :
        ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M6)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1336_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv)),Uw)) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1337_vebt__delete_Osimps_I3_J,axiom,
    ! [A: bool,B: bool,N2: nat] : vEBT_vebt_delete(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,N2))) = vEBT_Leaf(A,B) ).

% vebt_delete.simps(3)
tff(fact_1338_vebt__delete_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] : vEBT_vebt_delete(vEBT_Leaf(A,B),zero_zero_nat) = vEBT_Leaf(fFalse,B) ).

% vebt_delete.simps(1)
tff(fact_1339_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_1340_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv)),D))
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1341_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux)) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1342_frac__ge__0,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_ge_0
tff(fact_1343_frac__ge__0,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))) ).

% frac_ge_0
tff(fact_1344_frac__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real)) ).

% frac_lt_1
tff(fact_1345_frac__lt__1,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat)) ).

% frac_lt_1
tff(fact_1346_frac__1__eq,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X) ).

% frac_1_eq
tff(fact_1347_frac__1__eq,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).

% frac_1_eq
tff(fact_1348_invar__vebt_Ointros_I1_J,axiom,
    ! [A: bool,B: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).

% invar_vebt.intros(1)
tff(fact_1349_vebt__delete_Osimps_I2_J,axiom,
    ! [A: bool,B: bool] : vEBT_vebt_delete(vEBT_Leaf(A,B),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(A,fFalse) ).

% vebt_delete.simps(2)
tff(fact_1350_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_1351_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B3))
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B3),A)) ) ).

% ex_gt_or_lt
tff(fact_1352_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1353_dbl__inc__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_inc_def
tff(fact_1354_dbl__inc__def,axiom,
    ! [X: real] : aa_real_real(neg_nu8295874005876285629c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_inc_def
tff(fact_1355_dbl__inc__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_1356_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_1357_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_1358_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).

% List.finite_set
tff(fact_1359_List_Ofinite__set,axiom,
    ! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).

% List.finite_set
tff(fact_1360_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).

% List.finite_set
tff(fact_1361_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,set_Pr5648618587558075414at_nat(Xs))) ).

% List.finite_set
tff(fact_1362_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,set_Extended_enat2(Xs))) ).

% List.finite_set
tff(fact_1363_vebt__member_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Leaf(A,B)),X))
    <=> ( ( ( X = zero_zero_nat )
         => pp(A) )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => pp(B) )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_1364_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X3)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_1365_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),X),Y))
    <=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),X))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_1366_finite__has__maximal,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1367_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( member_real(X4,A2)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1368_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1369_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( finite_finite_rat(A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( member_rat(X4,A2)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1370_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( finite_finite_num(A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( member_num(X4,A2)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1371_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1372_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1373_finite__has__minimal,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1374_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( member_real(X4,A2)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1375_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1376_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( finite_finite_rat(A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( member_rat(X4,A2)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1377_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( finite_finite_num(A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( member_num(X4,A2)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1378_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1379_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1380_tvalid,axiom,
    pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),deg)) ).

% tvalid
tff(fact_1381_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1382_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1383_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1384_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
          & pp(aa_set_nat_bool(member_nat(N4),S3)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1385_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg))
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
             => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) )
          & ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1386_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M5) ).

% prod_decode_aux.cases
tff(fact_1387_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1388_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1389_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_1390_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2,Vb)),X))
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1391_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_1392_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_1393_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1394_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_1395_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: fun_set_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [A6: set_nat] :
            ( pp(aa_set_nat_bool(finite_finite_nat,A6))
           => ( ! [B6: set_nat] :
                  ( pp(aa_set_nat_bool(ord_less_set_nat(B6),A6))
                 => pp(aa_set_nat_bool(P,B6)) )
             => pp(aa_set_nat_bool(P,A6)) ) )
       => pp(aa_set_nat_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1396_finite__psubset__induct,axiom,
    ! [A2: set_int,P: fun_set_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [A6: set_int] :
            ( pp(aa_set_int_bool(finite_finite_int,A6))
           => ( ! [B6: set_int] :
                  ( pp(aa_set_int_bool(ord_less_set_int(B6),A6))
                 => pp(aa_set_int_bool(P,B6)) )
             => pp(aa_set_int_bool(P,A6)) ) )
       => pp(aa_set_int_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1397_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: fun_set_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [A6: set_complex] :
            ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A6))
           => ( ! [B6: set_complex] :
                  ( pp(aa_set_complex_bool(ord_less_set_complex(B6),A6))
                 => pp(aa_set_complex_bool(P,B6)) )
             => pp(aa_set_complex_bool(P,A6)) ) )
       => pp(aa_set_complex_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1398_finite__psubset__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ! [A6: set_Pr1261947904930325089at_nat] :
            ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A6))
           => ( ! [B6: set_Pr1261947904930325089at_nat] :
                  ( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(B6),A6))
                 => pp(aa_set3135850010578026034t_bool(P,B6)) )
             => pp(aa_set3135850010578026034t_bool(P,A6)) ) )
       => pp(aa_set3135850010578026034t_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1399_finite__psubset__induct,axiom,
    ! [A2: set_Extended_enat,P: fun_se2015444840911768724t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [A6: set_Extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A6))
           => ( ! [B6: set_Extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(B6),A6))
                 => pp(aa_set3347977610044113549t_bool(P,B6)) )
             => pp(aa_set3347977610044113549t_bool(P,A6)) ) )
       => pp(aa_set3347977610044113549t_bool(P,A2)) ) ) ).

% finite_psubset_induct
tff(fact_1400_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),B2))
    <=> ! [X3: complex] :
          ( pp(aa_set_complex_bool(member_complex(X3),set_complex2(Xs)))
         => pp(aa_set_complex_bool(member_complex(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_1401_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B2)
    <=> ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => member_real(X3,B2) ) ) ).

% subset_code(1)
tff(fact_1402_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),B2)
    <=> ! [X3: set_nat] :
          ( member_set_nat(X3,set_set_nat2(Xs))
         => member_set_nat(X3,B2) ) ) ).

% subset_code(1)
tff(fact_1403_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B2)
    <=> ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),B2)) ) ) ).

% subset_code(1)
tff(fact_1404_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B2))
    <=> ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_set_int_bool(member_int(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_1405_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),B2))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_set_nat_bool(member_nat(X3),B2)) ) ) ).

% subset_code(1)
tff(fact_1406_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( finite_finite_real(A2)
     => ( member_real(A,A2)
       => ? [X4: real] :
            ( member_real(X4,A2)
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1407_finite__has__maximal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),X4))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1408_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(A,A2)
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A2)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X4))
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1409_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( finite_finite_rat(A2)
     => ( member_rat(A,A2)
       => ? [X4: rat] :
            ( member_rat(X4,A2)
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X4))
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1410_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( finite_finite_num(A2)
     => ( member_num(A,A2)
       => ? [X4: num] :
            ( member_num(X4,A2)
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),X4))
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1411_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X4))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1412_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X4))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1413_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( finite_finite_real(A2)
     => ( member_real(A,A2)
       => ? [X4: real] :
            ( member_real(X4,A2)
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),A))
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1414_finite__has__minimal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),A2))
       => ? [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
            & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),A))
            & ! [Xa: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),A2))
               => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1415_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(A,A2)
       => ? [X4: set_nat] :
            ( member_set_nat(X4,A2)
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),A))
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1416_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( finite_finite_rat(A2)
     => ( member_rat(A,A2)
       => ? [X4: rat] :
            ( member_rat(X4,A2)
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),A))
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1417_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( finite_finite_num(A2)
     => ( member_num(A,A2)
       => ? [X4: num] :
            ( member_num(X4,A2)
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),A))
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1418_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(member_nat(A),A2))
       => ? [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),A))
            & ! [Xa: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa),A2))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1419_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_set_int_bool(member_int(A),A2))
       => ? [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),A))
            & ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),A2))
               => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X4))
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1420_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_1421_finite__list,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_1422_finite__list,axiom,
    ! [A2: set_int] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_1423_finite__list,axiom,
    ! [A2: set_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_1424_finite__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ? [Xs2: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(Xs2) = A2 ) ).

% finite_list
tff(fact_1425_finite__list,axiom,
    ! [A2: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ? [Xs2: list_Extended_enat] : set_Extended_enat2(Xs2) = A2 ) ).

% finite_list
tff(fact_1426_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M5))
         => ? [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N6))
              & pp(aa_set_nat_bool(member_nat(N6),S3)) ) )
     => ~ pp(aa_set_nat_bool(finite_finite_nat,S3)) ) ).

% unbounded_k_infinite
tff(fact_1427_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N4))
          & pp(aa_set_nat_bool(member_nat(N4),S3)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1428_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_1429_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( unique6319869463603278526ux_int(product_Pair_int_int(Q4,R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_1430_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(F,lattic8794016678065449205x_real(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1431_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X2)),aa_Ext2541572767299617110t_real(F,lattic1189837152898106425t_real(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1432_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( member_real(X2,S3)
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,lattic8440615504127631091l_real(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1433_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1434_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),S3))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,lattic2675449441010098035t_real(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1435_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1436_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X2)),aa_Extended_enat_rat(F,lattic3210252021154270693at_rat(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1437_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( member_real(X2,S3)
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1438_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1439_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),S3))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S3)))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1440_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S3))),aa_complex_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1441_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,lattic3210252021154270693at_rat(F,S3))),aa_Extended_enat_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1442_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( ( S3 != bot_bot_set_real )
       => ( member_real(Y,S3)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S3))),aa_real_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1443_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S3))),aa_nat_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1444_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S3))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S3))),aa_int_rat(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1445_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: fun_complex_num] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( S3 != bot_bot_set_complex )
       => ( pp(aa_set_complex_bool(member_complex(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S3))),aa_complex_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1446_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: fun_Ex8368542335153306515at_num] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Extended_enat_num(F,lattic402713867396545063at_num(F,S3))),aa_Extended_enat_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1447_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: fun_real_num] :
      ( finite_finite_real(S3)
     => ( ( S3 != bot_bot_set_real )
       => ( member_real(Y,S3)
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S3))),aa_real_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1448_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: fun_nat_num] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( S3 != bot_bot_set_nat )
       => ( pp(aa_set_nat_bool(member_nat(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_set_nat_nat(lattic4004264746738138117at_num(F),S3))),aa_nat_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1449_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: fun_int_num] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( S3 != bot_bot_set_int )
       => ( pp(aa_set_int_bool(member_int(Y),S3))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,lattic5003618458639192673nt_num(F,S3))),aa_int_num(F,Y))) ) ) ) ).

% arg_min_least
tff(fact_1450_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_1451_count__notin,axiom,
    ! [X: complex,Xs: list_complex] :
      ( ~ pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
     => ( aa_complex_nat(count_list_complex(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1452_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ member_real(X,set_real2(Xs))
     => ( aa_real_nat(count_list_real(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1453_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ member_set_nat(X,set_set_nat2(Xs))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1454_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1455_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => ( aa_nat_nat(count_list_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1456_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => ( aa_int_nat(count_list_int(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_1457_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1458_option_Osize__gen_I2_J,axiom,
    ! [X: fun_nat_nat,X22: nat] : size_option_nat(X,some_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1459_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X22: num] : size_option_num(X,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1460_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_1461_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) = N2 ) ) ).

% Suc_diff_1
tff(fact_1462_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1463_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1464_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1465_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1466_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1467_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2) )
    <=> ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1468_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K) )
    <=> ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1469_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1470_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
    <=> ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1471_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_1472_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1473_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_1474_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_1475_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_1476_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1477_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_1478_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_1479_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_1480_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1481_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_1482_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_1483_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_1484_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1485_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_1486_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_1487_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_1488_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1489_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_1490_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_1491_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_1492_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_1493_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_1494_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_1495_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_1496_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_1497_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_1498_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_1499_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_1500_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_1501_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_1502_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_1503_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1504_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_1505_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_1506_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_1507_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_1508_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_1509_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_1510_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_1511_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_1512_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_1513_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1514_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_1515_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_1516_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_1517_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_1518_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_1519_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_1520_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_1521_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_1522_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_1523_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_1524_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_1525_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_1526_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_1527_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_1528_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_1529_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_1530_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_1531_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_1532_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_1533_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_1534_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_1535_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_1536_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_1537_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_1538_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1539_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1540_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% mult_less_cancel2
tff(fact_1541_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% nat_0_less_mult_iff
tff(fact_1542_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ).

% mult_Suc_right
tff(fact_1543_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),K) ).

% Suc_diff_diff
tff(fact_1544_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% diff_Suc_Suc
tff(fact_1545_diff__0__eq__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N2) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1546_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1547_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1548_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_1549_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> vEBT_VEBT_lesseq(some_nat(X),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_1550_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_1551_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_1552_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% diff_ge_0_iff_ge
tff(fact_1553_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_1554_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_1555_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).

% diff_gt_0_iff_gt
tff(fact_1556_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_1557_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_1558_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1559_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_1560_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_1561_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_1562_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1563_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_1564_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_1565_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_1566_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1567_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_1568_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_1569_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_1570_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1571_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_1572_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1573_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1574_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1575_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_1576_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1577_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1578_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1579_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_1580_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_1581_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_1582_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_1583_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1584_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_1585_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) ) ) ).

% one_le_mult_iff
tff(fact_1586_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% mult_le_cancel2
tff(fact_1587_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% zero_less_diff
tff(fact_1588_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% diff_is_0_eq
tff(fact_1589_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1590_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1591_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1592_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1593_diff__Suc__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N2)),one_one_nat) = N2 ).

% diff_Suc_1
tff(fact_1594_Suc__pred,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))) = N2 ) ) ).

% Suc_pred
tff(fact_1595_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1596_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1597_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_1598_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_1599_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_1600_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_1601_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_1602_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_1603_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_1604_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_1605_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_1606_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_1607_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_1608_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_1609_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_1610_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_1611_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_1612_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_1613_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_1614_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_1615_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_1616_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1617_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_1618_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_1619_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_1620_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_1621_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_1622_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_1623_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_1624_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_1625_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_1626_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_1627_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_1628_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_1629_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_1630_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_1631_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_1632_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_1633_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_1634_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_1635_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)) ).

% diff_mult_distrib
tff(fact_1636_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) ).

% diff_mult_distrib2
tff(fact_1637_inf__period_I2_J,axiom,
    ! [P: fun_real_bool,D4: real,Q: fun_real_bool] :
      ( ! [X4: real,K2: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
     => ( ! [X4: real,K2: real] :
            ( pp(aa_real_bool(Q,X4))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
       => ! [X2: real,K4: real] :
            ( ( pp(aa_real_bool(P,X2))
              | pp(aa_real_bool(Q,X2)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))))
              | pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_1638_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D4: rat,Q: fun_rat_bool] :
      ( ! [X4: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X4))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D4)))) )
     => ( ! [X4: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X4))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D4)))) )
       => ! [X2: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X2))
              | pp(aa_rat_bool(Q,X2)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D4))))
              | pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_1639_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D4: int,Q: fun_int_bool] :
      ( ! [X4: int,K2: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(Q,X4))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
       => ! [X2: int,K4: int] :
            ( ( pp(aa_int_bool(P,X2))
              | pp(aa_int_bool(Q,X2)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))))
              | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))) ) ) ) ) ).

% inf_period(2)
tff(fact_1640_inf__period_I1_J,axiom,
    ! [P: fun_real_bool,D4: real,Q: fun_real_bool] :
      ( ! [X4: real,K2: real] :
          ( pp(aa_real_bool(P,X4))
        <=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
     => ( ! [X4: real,K2: real] :
            ( pp(aa_real_bool(Q,X4))
          <=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D4)))) )
       => ! [X2: real,K4: real] :
            ( ( pp(aa_real_bool(P,X2))
              & pp(aa_real_bool(Q,X2)) )
          <=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))))
              & pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_1641_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D4: rat,Q: fun_rat_bool] :
      ( ! [X4: rat,K2: rat] :
          ( pp(aa_rat_bool(P,X4))
        <=> pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D4)))) )
     => ( ! [X4: rat,K2: rat] :
            ( pp(aa_rat_bool(Q,X4))
          <=> pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K2),D4)))) )
       => ! [X2: rat,K4: rat] :
            ( ( pp(aa_rat_bool(P,X2))
              & pp(aa_rat_bool(Q,X2)) )
          <=> ( pp(aa_rat_bool(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D4))))
              & pp(aa_rat_bool(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_1642_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D4: int,Q: fun_int_bool] :
      ( ! [X4: int,K2: int] :
          ( pp(aa_int_bool(P,X4))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(Q,X4))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
       => ! [X2: int,K4: int] :
            ( ( pp(aa_int_bool(P,X2))
              & pp(aa_int_bool(Q,X2)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))))
              & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))) ) ) ) ) ).

% inf_period(1)
tff(fact_1643_square__diff__square__factored,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).

% square_diff_square_factored
tff(fact_1644_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_rat_rat(minus_minus_rat(X),Y)) ).

% square_diff_square_factored
tff(fact_1645_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_1646_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_1647_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_1648_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_1649_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1650_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1651_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_1652_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A)),B)) ).

% mult_diff_mult
tff(fact_1653_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),A)),B)) ).

% mult_diff_mult
tff(fact_1654_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A)),B)) ).

% mult_diff_mult
tff(fact_1655_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1656_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1657_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1658_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1659_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1660_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D)) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1661_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_1662_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_1663_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D))) ) ).

% less_add_iff2
tff(fact_1664_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_1665_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_1666_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C)),D)) ) ).

% less_add_iff1
tff(fact_1667_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_1668_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ).

% square_diff_one_factored
tff(fact_1669_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_1670_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_1671_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) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1672_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1673_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) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1674_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_right_mono
tff(fact_1675_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_right_mono
tff(fact_1676_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_right_mono
tff(fact_1677_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_left_mono
tff(fact_1678_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_left_mono
tff(fact_1679_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_left_mono
tff(fact_1680_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_mono
tff(fact_1681_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_mono
tff(fact_1682_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_mono
tff(fact_1683_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_1684_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1685_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_1686_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).

% diff_strict_right_mono
tff(fact_1687_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C))) ) ).

% diff_strict_right_mono
tff(fact_1688_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).

% diff_strict_right_mono
tff(fact_1689_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).

% diff_strict_left_mono
tff(fact_1690_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B))) ) ).

% diff_strict_left_mono
tff(fact_1691_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).

% diff_strict_left_mono
tff(fact_1692_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) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1693_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1694_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) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).

% diff_eq_diff_less
tff(fact_1695_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).

% diff_strict_mono
tff(fact_1696_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D))) ) ) ).

% diff_strict_mono
tff(fact_1697_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).

% diff_strict_mono
tff(fact_1698_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_1699_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_1700_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_1701_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_1702_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_1703_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_1704_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_1705_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_1706_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_1707_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_1708_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_1709_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_1710_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_add_eq
tff(fact_1711_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_1712_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_1713_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_diff_eq2
tff(fact_1714_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_1715_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_1716_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).

% add_diff_eq
tff(fact_1717_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_1718_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_1719_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_1720_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_1721_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_1722_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_1723_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_1724_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1725_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1726_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_1727_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_1728_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_1729_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_1730_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_1731_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1732_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_1733_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_1734_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_1735_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1736_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_1737_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_1738_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_1739_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1740_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_1741_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_1742_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_1743_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1744_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_1745_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_1746_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_1747_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1748_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_1749_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_1750_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : X != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1751_zero__induct__lemma,axiom,
    ! [P: fun_nat_bool,K: nat,I: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(P,aa_nat_nat(suc,N)))
           => pp(aa_nat_bool(P,N)) )
       => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).

% zero_induct_lemma
tff(fact_1752_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1753_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1754_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1755_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1756_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_1757_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_1758_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_1759_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_1760_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_1761_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_1762_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_1763_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_1764_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_1765_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1766_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_1767_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_1768_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_1769_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1770_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_1771_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_1772_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1773_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_1774_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_1775_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_1776_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_1777_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_1778_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_1779_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Y)),aa_rat_rat(times_times_rat(X),Z2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W2),Z2)),aa_rat_rat(times_times_rat(X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_1780_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W2),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_1781_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W2 = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_1782_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1783_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_1784_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_1785_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_1786_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_1787_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_1788_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_1789_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1790_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_1791_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_1792_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_1793_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M) = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
tff(fact_1794_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N2)),K)) ) ).

% less_imp_diff_less
tff(fact_1795_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_1796_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N2) )
    <=> ( M = N2 ) ) ).

% Suc_mult_cancel1
tff(fact_1797_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).

% diff_le_mono2
tff(fact_1798_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).

% le_diff_iff'
tff(fact_1799_diff__le__self,axiom,
    ! [M: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),M)) ).

% diff_le_self
tff(fact_1800_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),L))) ) ).

% diff_le_mono
tff(fact_1801_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1802_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ) ).

% le_diff_iff
tff(fact_1803_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K) )
        <=> ( M = N2 ) ) ) ) ).

% eq_diff_iff
tff(fact_1804_mult__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N2) = zero_zero_nat ).

% mult_0
tff(fact_1805_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),N2) = M ).

% diff_add_inverse2
tff(fact_1806_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)),N2) = M ).

% diff_add_inverse
tff(fact_1807_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% diff_cancel2
tff(fact_1808_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ).

% Nat.diff_cancel
tff(fact_1809_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ).

% mult_le_mono2
tff(fact_1810_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ).

% mult_le_mono1
tff(fact_1811_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L))) ) ) ).

% mult_le_mono
tff(fact_1812_le__square,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_square
tff(fact_1813_le__cube,axiom,
    ! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)))) ).

% le_cube
tff(fact_1814_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) ).

% add_mult_distrib2
tff(fact_1815_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)) ).

% add_mult_distrib
tff(fact_1816_nat__mult__1__right,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),one_one_nat) = N2 ).

% nat_mult_1_right
tff(fact_1817_nat__mult__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N2) = N2 ).

% nat_mult_1
tff(fact_1818_mult__eq__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = zero_zero_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N2)) ) ) ) ).

% mult_eq_if
tff(fact_1819_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% le_iff_diff_le_0
tff(fact_1820_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_1821_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_1822_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).

% less_iff_diff_less_0
tff(fact_1823_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_1824_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_1825_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1826_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1827_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1828_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1829_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1830_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1831_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1832_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1833_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A))) ) ).

% le_add_diff
tff(fact_1834_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_1835_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% le_diff_eq
tff(fact_1836_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C)) ) ).

% le_diff_eq
tff(fact_1837_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% le_diff_eq
tff(fact_1838_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_le_eq
tff(fact_1839_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% diff_le_eq
tff(fact_1840_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_le_eq
tff(fact_1841_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N2: real,J: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N2),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1842_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N2: rat,J: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N2),aa_rat_rat(plus_plus_rat(J),K)))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N2),aa_rat_rat(plus_plus_rat(J),K)))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(N2),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1843_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N2: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1844_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N2: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N2),K)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1845_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N2),K))) ) ).

% add_le_imp_le_diff
tff(fact_1846_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),N2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(N2),K))) ) ).

% add_le_imp_le_diff
tff(fact_1847_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ).

% add_le_imp_le_diff
tff(fact_1848_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N2),K))) ) ).

% add_le_imp_le_diff
tff(fact_1849_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).

% less_diff_eq
tff(fact_1850_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C)) ) ).

% less_diff_eq
tff(fact_1851_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).

% less_diff_eq
tff(fact_1852_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).

% diff_less_eq
tff(fact_1853_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B))) ) ).

% diff_less_eq
tff(fact_1854_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).

% diff_less_eq
tff(fact_1855_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1856_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1857_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1858_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1859_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1860_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1861_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1862_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1863_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_mult_iff
tff(fact_1864_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_1865_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_1866_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1867_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1868_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1869_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1870_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonpos_nonneg
tff(fact_1871_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_1872_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_1873_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_1874_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_nonneg_nonpos
tff(fact_1875_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_1876_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_1877_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_1878_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_1879_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_1880_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_1881_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_nonneg_nonneg
tff(fact_1882_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ).

% split_mult_neg_le
tff(fact_1883_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_1884_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ).

% split_mult_neg_le
tff(fact_1885_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_1886_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_1887_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_1888_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).

% mult_le_0_iff
tff(fact_1889_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono
tff(fact_1890_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono
tff(fact_1891_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_right_mono
tff(fact_1892_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono
tff(fact_1893_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_1894_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_1895_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_right_mono_neg
tff(fact_1896_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono
tff(fact_1897_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono
tff(fact_1898_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_left_mono
tff(fact_1899_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono
tff(fact_1900_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_1901_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_1902_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_nonpos_nonpos
tff(fact_1903_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_1904_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_1905_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_left_mono_neg
tff(fact_1906_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ).

% split_mult_pos_le
tff(fact_1907_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ).

% split_mult_pos_le
tff(fact_1908_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).

% split_mult_pos_le
tff(fact_1909_zero__le__square,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A))) ).

% zero_le_square
tff(fact_1910_zero__le__square,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A))) ).

% zero_le_square
tff(fact_1911_zero__le__square,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A))) ).

% zero_le_square
tff(fact_1912_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_1913_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_1914_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_1915_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono'
tff(fact_1916_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_1917_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_mono
tff(fact_1918_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_1919_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_mono
tff(fact_1920_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1921_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1922_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1923_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1924_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1925_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1926_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1927_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_1928_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_1929_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_1930_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono
tff(fact_1931_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1932_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1933_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1934_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1935_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1936_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1937_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1938_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1939_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1940_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono
tff(fact_1941_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1942_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1943_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1944_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1945_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1946_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1947_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1948_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1949_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1950_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_1951_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_1952_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_1953_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos2
tff(fact_1954_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).

% zero_less_mult_pos
tff(fact_1955_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_1956_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).

% zero_less_mult_pos
tff(fact_1957_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).

% zero_less_mult_pos
tff(fact_1958_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_mult_iff
tff(fact_1959_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_1960_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_1961_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).

% mult_pos_neg2
tff(fact_1962_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_1963_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_1964_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_1965_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_pos_pos
tff(fact_1966_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_pos_pos
tff(fact_1967_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).

% mult_pos_pos
tff(fact_1968_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_pos_pos
tff(fact_1969_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_pos_neg
tff(fact_1970_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_1971_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_1972_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_1973_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).

% mult_neg_pos
tff(fact_1974_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_1975_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_1976_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_1977_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_1978_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_1979_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% mult_less_0_iff
tff(fact_1980_not__square__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real)) ).

% not_square_less_zero
tff(fact_1981_not__square__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat)) ).

% not_square_less_zero
tff(fact_1982_not__square__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int)) ).

% not_square_less_zero
tff(fact_1983_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).

% mult_neg_neg
tff(fact_1984_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).

% mult_neg_neg
tff(fact_1985_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).

% mult_neg_neg
tff(fact_1986_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_1987_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1988_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_1989_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_1990_less__1__mult,axiom,
    ! [M: real,N2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),N2))) ) ) ).

% less_1_mult
tff(fact_1991_less__1__mult,axiom,
    ! [M: rat,N2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),N2))) ) ) ).

% less_1_mult
tff(fact_1992_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2))) ) ) ).

% less_1_mult
tff(fact_1993_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N2))) ) ) ).

% less_1_mult
tff(fact_1994_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),aa_nat_nat(suc,M))) ).

% diff_less_Suc
tff(fact_1995_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2) ) ) ).

% Suc_diff_Suc
tff(fact_1996_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),M)) ) ) ).

% diff_less
tff(fact_1997_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) ) ).

% Suc_diff_le
tff(fact_1998_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).

% diff_less_mono
tff(fact_1999_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ) ).

% less_diff_iff
tff(fact_2000_set__encode__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_set_nat_bool(finite_finite_nat,B2))
       => ( ( aa_set_nat_nat(nat_set_encode,A2) = aa_set_nat_nat(nat_set_encode,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_2001_diff__add__0,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_2002_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_2003_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).

% less_diff_conv
tff(fact_2004_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% Suc_mult_less_cancel1
tff(fact_2005_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_2006_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_2007_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_2008_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_2009_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).

% le_diff_conv
tff(fact_2010_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K))) ) ) ).

% mult_less_mono1
tff(fact_2011_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J))) ) ) ).

% mult_less_mono2
tff(fact_2012_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% Suc_mult_le_cancel1
tff(fact_2013_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N2) ).

% diff_Suc_eq_diff_pred
tff(fact_2014_mult__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) ).

% mult_Suc
tff(fact_2015_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2016_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,D5: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),D5)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_2017_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2018_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2019_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2020_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2021_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2022_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2023_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2024_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2025_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2026_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2027_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2028_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_right_le_imp_le
tff(fact_2029_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2030_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2031_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2032_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_left_le_imp_le
tff(fact_2033_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2034_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2035_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2036_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2037_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2038_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2039_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2040_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2041_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_right
tff(fact_2042_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2043_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2044_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2045_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2046_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2047_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2048_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2049_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_right_less_imp_less
tff(fact_2050_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2051_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2052_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).

% mult_less_cancel_left
tff(fact_2053_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2054_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2055_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2056_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).

% mult_strict_mono
tff(fact_2057_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2058_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2059_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2060_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% mult_left_less_imp_less
tff(fact_2061_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2062_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2063_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_right
tff(fact_2064_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2065_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2066_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).

% mult_le_cancel_left
tff(fact_2067_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2068_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2069_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_2070_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2071_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2072_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_2073_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2074_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2075_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_2076_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real)) ) ) ) ).

% mult_le_one
tff(fact_2077_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_2078_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_2079_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_2080_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2081_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2082_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2083_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).

% mult_left_le
tff(fact_2084_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).

% not_sum_squares_lt_zero
tff(fact_2085_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)) ).

% not_sum_squares_lt_zero
tff(fact_2086_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_2087_diff__Suc__less,axiom,
    ! [N2: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,I))),N2)) ) ).

% diff_Suc_less
tff(fact_2088_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D6: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
              & ~ pp(aa_nat_bool(P,D6)) ) ) ) ).

% nat_diff_split_asm
tff(fact_2089_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A: nat,B: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D6: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D6) )
           => pp(aa_nat_bool(P,D6)) ) ) ) ).

% nat_diff_split
tff(fact_2090_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).

% less_diff_conv2
tff(fact_2091_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M))) ) ) ).

% n_less_n_mult_m
tff(fact_2092_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2))) ) ) ).

% n_less_m_mult_n
tff(fact_2093_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2))) ) ) ).

% one_less_mult
tff(fact_2094_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_nat(nat_set_encode,A2) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_2095_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2096_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2097_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2098_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2099_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2100_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2101_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2102_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2103_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2104_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2105_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2106_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2107_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2108_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2109_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2110_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2111_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2112_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2113_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2114_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2115_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2116_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2117_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2118_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2119_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X)),Y)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2120_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),one_one_rat))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X)),Y)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_2121_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2122_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2123_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2124_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_2125_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( N2 = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_2126_add__eq__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = N2 ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N2)) ) ) ) ).

% add_eq_if
tff(fact_2127_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2128_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real))
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real) ) ) ) ).

% frac_add
tff(fact_2129_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat))
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat) ) ) ) ).

% frac_add
tff(fact_2130_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2131_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2132_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2133_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2134_succ__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_correct
tff(fact_2135_pred__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).

% pred_correct
tff(fact_2136_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(vEBT_VEBT_less(some_nat(X),some_nat(Y))) ) ).

% less_shift
tff(fact_2137_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
    <=> pp(vEBT_VEBT_greater(some_nat(X),some_nat(Y))) ) ).

% greater_shift
tff(fact_2138_succ__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Sx)) ) ) ).

% succ_corr
tff(fact_2139_pred__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Px: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Px) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Px)) ) ) ).

% pred_corr
tff(fact_2140_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_set_nat_bool(ord_less_set_nat(A2),B2)) ) ) ).

% psubsetI
tff(fact_2141_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2142_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
     => ? [B3: complex] : pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_2143_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ord_less_set_real(A2,B2)
     => ? [B3: real] : member_real(B3,minus_minus_set_real(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2144_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ord_less_set_set_nat(A2,B2)
     => ? [B3: set_nat] : member_set_nat(B3,minus_2163939370556025621et_nat(B2,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_2145_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ? [B3: int] : pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_2146_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => ? [B3: nat] : pp(aa_set_nat_bool(member_nat(B3),minus_minus_set_nat(B2,A2))) ) ).

% psubset_imp_ex_mem
tff(fact_2147_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2148_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2149_psubsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
     => ( pp(aa_set_complex_bool(member_complex(C),A2))
       => pp(aa_set_complex_bool(member_complex(C),B2)) ) ) ).

% psubsetD
tff(fact_2150_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ord_less_set_real(A2,B2)
     => ( member_real(C,A2)
       => member_real(C,B2) ) ) ).

% psubsetD
tff(fact_2151_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ord_less_set_set_nat(A2,B2)
     => ( member_set_nat(C,A2)
       => member_set_nat(C,B2) ) ) ).

% psubsetD
tff(fact_2152_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => ( pp(aa_set_nat_bool(member_nat(C),A2))
       => pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).

% psubsetD
tff(fact_2153_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
     => ( pp(aa_set_int_bool(member_int(C),A2))
       => pp(aa_set_int_bool(member_int(C),B2)) ) ) ).

% psubsetD
tff(fact_2154_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2) )
      <=> ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2155_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% nat_mult_less_cancel1
tff(fact_2156_not__psubset__empty,axiom,
    ! [A2: set_Extended_enat] : ~ pp(aa_set3347977610044113549t_bool(ord_le2529575680413868914d_enat(A2),bot_bo7653980558646680370d_enat)) ).

% not_psubset_empty
tff(fact_2157_not__psubset__empty,axiom,
    ! [A2: set_real] : ~ ord_less_set_real(A2,bot_bot_set_real) ).

% not_psubset_empty
tff(fact_2158_not__psubset__empty,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(A2),bot_bot_set_nat)) ).

% not_psubset_empty
tff(fact_2159_not__psubset__empty,axiom,
    ! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).

% not_psubset_empty
tff(fact_2160_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => ~ ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2)) ) ) ).

% psubsetE
tff(fact_2161_psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_2162_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2)) ) ).

% psubset_imp_subset
tff(fact_2163_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
     => ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
       => pp(aa_set_nat_bool(ord_less_set_nat(A2),C4)) ) ) ).

% psubset_subset_trans
tff(fact_2164_subset__not__subset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
        & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2)) ) ) ).

% subset_not_subset_eq
tff(fact_2165_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
     => ( pp(aa_set_nat_bool(ord_less_set_nat(B2),C4))
       => pp(aa_set_nat_bool(ord_less_set_nat(A2),C4)) ) ) ).

% subset_psubset_trans
tff(fact_2166_subset__iff__psubset__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
    <=> ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2167_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% nat_mult_le_cancel1
tff(fact_2168_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N2 ) ) ) ).

% nat_eq_add_iff1
tff(fact_2169_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N2) ) ) ) ).

% nat_eq_add_iff2
tff(fact_2170_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N2)) ) ) ).

% nat_le_add_iff1
tff(fact_2171_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N2))) ) ) ).

% nat_le_add_iff2
tff(fact_2172_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N2) ) ) ).

% nat_diff_add_eq1
tff(fact_2173_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N2)) ) ) ).

% nat_diff_add_eq2
tff(fact_2174_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2175_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2176_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2177_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2178_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y),Y))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2179_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2180_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N2)) ) ) ).

% nat_less_add_iff1
tff(fact_2181_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N2))) ) ) ).

% nat_less_add_iff2
tff(fact_2182_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Ma),X))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_2183_mul__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = Z2 )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X)),some_nat(Y)) = some_nat(Z2) ) ) ).

% mul_shift
tff(fact_2184_add__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = Z2 )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X)),some_nat(Y)) = some_nat(Z2) ) ) ).

% add_shift
tff(fact_2185_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_maxt(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% maxt_corr
tff(fact_2186_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_maxt(T) = some_nat(X) ) ) ) ).

% maxt_sound
tff(fact_2187_mint__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_mint(T) = some_nat(X) )
       => pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).

% mint_corr
tff(fact_2188_mint__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X))
       => ( vEBT_vebt_mint(T) = some_nat(X) ) ) ) ).

% mint_sound
tff(fact_2189_mult__le__cancel__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2190_mult__le__cancel__iff1,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(Y),Z2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2191_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_2192_mult__le__cancel__iff2,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2193_mult__le__cancel__iff2,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),X)),aa_rat_rat(times_times_rat(Z2),Y)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2194_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Y)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_2195_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(X) )
     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X)) ) ).

% maxbmo
tff(fact_2196_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).

% minminNull
tff(fact_2197_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_2198_mint__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% mint_member
tff(fact_2199_maxt__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).

% maxt_member
tff(fact_2200_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Mini: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_2201_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_2202_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_2203_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_2204_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_2205_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_2206_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N2: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N2)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_2207_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : vEBT_vebt_pred(vEBT_Leaf(Uu,Uv),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_2208_option_Osize__gen_I1_J,axiom,
    ! [X: fun_nat_nat] : size_option_nat(X,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_2209_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_2210_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_2211_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_2212_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_2213_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_2214_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_2215_vebt__pred_Osimps_I2_J,axiom,
    ! [A: bool,Uw: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw),aa_nat_nat(suc,zero_zero_nat)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,Uw),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).

% vebt_pred.simps(2)
tff(fact_2216_vebt__succ_Osimps_I1_J,axiom,
    ! [B: bool,Uu: bool] :
      ( ( pp(B)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B),zero_zero_nat) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B),zero_zero_nat) = none_nat ) ) ) ).

% vebt_succ.simps(1)
tff(fact_2217_vebt__pred_Osimps_I3_J,axiom,
    ! [B: bool,A: bool,Va2: nat] :
      ( ( pp(B)
       => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_2218_mult__less__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_2219_mult__less__iff1,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(Y),Z2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_2220_mult__less__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_2221_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: bool,A: bool] :
      ( ( pp(B)
       => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
      & ( ~ pp(B)
       => ( ( pp(A)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
          & ( ~ pp(A)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_2222_vebt__mint_Osimps_I1_J,axiom,
    ! [A: bool,B: bool] :
      ( ( pp(A)
       => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
      & ( ~ pp(A)
       => ( ( pp(B)
           => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
          & ( ~ pp(B)
           => ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_2223_diff__shunt__var,axiom,
    ! [X: set_Extended_enat,Y: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(X,Y) = bot_bo7653980558646680370d_enat )
    <=> pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(X),Y)) ) ).

% diff_shunt_var
tff(fact_2224_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
    <=> ord_less_eq_set_real(X,Y) ) ).

% diff_shunt_var
tff(fact_2225_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).

% diff_shunt_var
tff(fact_2226_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% diff_shunt_var
tff(fact_2227_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
       => ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_2228_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
         => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_2229_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),zero_zero_nat)
     => ( ! [A3: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B3: bool,N: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,N)))
         => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2),X4)
             => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),X4)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),X4) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_2230_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),X4)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),X4)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_2231_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,Uv2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)
     => ( ! [A3: bool,Uw2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,Uw2),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: bool,B3: bool,Va: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Vf2)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),X4) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_2232_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: bool,B3: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B3),zero_zero_nat)
     => ( ! [Uv2: bool,Uw2: bool,N: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve2)
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),X4) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_2233_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary),Uu) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary) ).

% vebt_delete.simps(4)
tff(fact_2234_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_2235_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_2236_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).

% VEBT_internal.membermima.simps(2)
tff(fact_2237_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2238_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2239_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( pp(A3)
                 => ( Y = some_nat(zero_zero_nat) ) )
                & ( ~ pp(A3)
                 => ( ( pp(B3)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_2240_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( pp(B3)
                 => ( Y = some_nat(one_one_nat) ) )
                & ( ~ pp(B3)
                 => ( ( pp(A3)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_2241_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X4)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] : X != 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] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_2242_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: bool,B3: bool,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),X4)
     => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts,S2),X4)
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2),X4)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2),X4)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_2243_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( pp(B3)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( ( pp(A3)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf(A3,B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_2244_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( pp(A3)
                   => ( Y = some_nat(zero_zero_nat) ) )
                  & ( ~ pp(A3)
                   => ( ( pp(B3)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( Y = none_nat ) ) ) ) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf(A3,B3))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi2) )
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_2245_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2246_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2247_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2248_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2249_option_Osize_I4_J,axiom,
    ! [X22: nat] : size_size_option_nat(some_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2250_option_Osize_I4_J,axiom,
    ! [X22: num] : size_size_option_num(some_num(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2251_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_2252_frac__unique__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = A )
    <=> ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A),ring_1_Ints_real)
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) ) ) ).

% frac_unique_iff
tff(fact_2253_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = A )
    <=> ( member_rat(aa_rat_rat(minus_minus_rat(X),A),ring_1_Ints_rat)
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) ) ) ).

% frac_unique_iff
tff(fact_2254_dbl__dec__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_dec_def
tff(fact_2255_dbl__dec__def,axiom,
    ! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_dec_def
tff(fact_2256_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_2257_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_2258_prod__encode__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(nat_prod_encode,X) = aa_Pro4243173770044397973at_nat(nat_prod_encode,Y) )
    <=> ( X = Y ) ) ).

% prod_encode_eq
tff(fact_2259_frac__in__Ints__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(archim2898591450579166408c_real,X),ring_1_Ints_real)
    <=> member_real(X,ring_1_Ints_real) ) ).

% frac_in_Ints_iff
tff(fact_2260_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_2261_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_2262_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_2263_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_2264_frac__eq__0__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = zero_zero_real )
    <=> member_real(X,ring_1_Ints_real) ) ).

% frac_eq_0_iff
tff(fact_2265_frac__eq__0__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = zero_zero_rat )
    <=> member_rat(X,ring_1_Ints_rat) ) ).

% frac_eq_0_iff
tff(fact_2266_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X)))
    <=> ~ member_real(X,ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_2267_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)))
    <=> ~ member_rat(X,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_2268_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( size_size_char(X) != size_size_char(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2269_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2270_size__neq__size__imp__neq,axiom,
    ! [X: list_bool,Y: list_bool] :
      ( ( size_size_list_bool(X) != size_size_list_bool(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2271_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X) != size_size_list_nat(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2272_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2273_Ints__0,axiom,
    pp(aa_set_complex_bool(member_complex(zero_zero_complex),ring_1_Ints_complex)) ).

% Ints_0
tff(fact_2274_Ints__0,axiom,
    member_real(zero_zero_real,ring_1_Ints_real) ).

% Ints_0
tff(fact_2275_Ints__0,axiom,
    member_rat(zero_zero_rat,ring_1_Ints_rat) ).

% Ints_0
tff(fact_2276_Ints__0,axiom,
    pp(aa_set_int_bool(member_int(zero_zero_int),ring_1_Ints_int)) ).

% Ints_0
tff(fact_2277_Ints__mult,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_set_complex_bool(member_complex(A),ring_1_Ints_complex))
     => ( pp(aa_set_complex_bool(member_complex(B),ring_1_Ints_complex))
       => pp(aa_set_complex_bool(member_complex(aa_complex_complex(times_times_complex(A),B)),ring_1_Ints_complex)) ) ) ).

% Ints_mult
tff(fact_2278_Ints__mult,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_mult
tff(fact_2279_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(times_times_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_mult
tff(fact_2280_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => ( pp(aa_set_int_bool(member_int(B),ring_1_Ints_int))
       => pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),ring_1_Ints_int)) ) ) ).

% Ints_mult
tff(fact_2281_Ints__add,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_set_complex_bool(member_complex(A),ring_1_Ints_complex))
     => ( pp(aa_set_complex_bool(member_complex(B),ring_1_Ints_complex))
       => pp(aa_set_complex_bool(member_complex(aa_complex_complex(plus_plus_complex(A),B)),ring_1_Ints_complex)) ) ) ).

% Ints_add
tff(fact_2282_Ints__add,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_add
tff(fact_2283_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(plus_plus_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_add
tff(fact_2284_Ints__add,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => ( pp(aa_set_int_bool(member_int(B),ring_1_Ints_int))
       => pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),ring_1_Ints_int)) ) ) ).

% Ints_add
tff(fact_2285_Ints__1,axiom,
    member_rat(one_one_rat,ring_1_Ints_rat) ).

% Ints_1
tff(fact_2286_Ints__1,axiom,
    pp(aa_set_int_bool(member_int(one_one_int),ring_1_Ints_int)) ).

% Ints_1
tff(fact_2287_Ints__1,axiom,
    member_real(one_one_real,ring_1_Ints_real) ).

% Ints_1
tff(fact_2288_Ints__1,axiom,
    pp(aa_set_complex_bool(member_complex(one_one_complex),ring_1_Ints_complex)) ).

% Ints_1
tff(fact_2289_Ints__diff,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_set_complex_bool(member_complex(A),ring_1_Ints_complex))
     => ( pp(aa_set_complex_bool(member_complex(B),ring_1_Ints_complex))
       => pp(aa_set_complex_bool(member_complex(aa_complex_complex(minus_minus_complex(A),B)),ring_1_Ints_complex)) ) ) ).

% Ints_diff
tff(fact_2290_Ints__diff,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B),ring_1_Ints_real) ) ) ).

% Ints_diff
tff(fact_2291_Ints__diff,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(minus_minus_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_diff
tff(fact_2292_Ints__diff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => ( pp(aa_set_int_bool(member_int(B),ring_1_Ints_int))
       => pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),ring_1_Ints_int)) ) ) ).

% Ints_diff
tff(fact_2293_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( pp(aa_set_complex_bool(member_complex(A),ring_1_Ints_complex))
     => ( ( aa_complex_complex(plus_plus_complex(A),A) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2294_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2295_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2296_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2297_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( pp(aa_set_complex_bool(member_complex(A),ring_1_Ints_complex))
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),A)),A) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_2298_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_2299_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_2300_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_2301_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_1
tff(fact_2302_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).

% le_prod_encode_2
tff(fact_2303_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% Ints_odd_less_0
tff(fact_2304_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),A)),A)),zero_zero_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% Ints_odd_less_0
tff(fact_2305_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% Ints_odd_less_0
tff(fact_2306_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
      <=> pp(Y) )
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y)
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y)
                   => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( pp(Y)
                     => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ pp(Y)
                       => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_2307_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_2308_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
     => ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2))) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_2309_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2,Summary) ).

% vebt_insert.simps(4)
tff(fact_2310_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2311_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2312_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2313_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M)) ) ) ) ).

% power_decreasing_iff
tff(fact_2314_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_2315_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z2)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2)) ) ).

% pair_less_iff1
tff(fact_2316_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_2317_frac__neg,axiom,
    ! [X: real] :
      ( ( member_real(X,ring_1_Ints_real)
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = zero_zero_real ) )
      & ( ~ member_real(X,ring_1_Ints_real)
       => ( aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(archim2898591450579166408c_real,X)) ) ) ) ).

% frac_neg
tff(fact_2318_frac__neg,axiom,
    ! [X: rat] :
      ( ( member_rat(X,ring_1_Ints_rat)
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = zero_zero_rat ) )
      & ( ~ member_rat(X,ring_1_Ints_rat)
       => ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(archimedean_frac_rat,X)) ) ) ) ).

% frac_neg
tff(fact_2319_power__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Y) = Z2 )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X)),some_nat(Y)) = some_nat(Z2) ) ) ).

% power_shift
tff(fact_2320_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_2321_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_2322_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_2323_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_2324_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_2325_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_2326_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_2327_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_2328_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_2329_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_2330_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_2331_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_2332_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_2333_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).

% verit_minus_simplify(4)
tff(fact_2334_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).

% verit_minus_simplify(4)
tff(fact_2335_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_2336_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ).

% compl_le_compl_iff
tff(fact_2337_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% neg_le_iff_le
tff(fact_2338_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% neg_le_iff_le
tff(fact_2339_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% neg_le_iff_le
tff(fact_2340_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% neg_le_iff_le
tff(fact_2341_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_2342_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_2343_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_2344_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_2345_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_2346_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_2347_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_2348_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_2349_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_2350_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_2351_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_2352_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_2353_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_2354_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_2355_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_2356_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_2357_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_2358_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_2359_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_2360_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_2361_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_2362_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_2363_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_2364_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% neg_less_iff_less
tff(fact_2365_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% neg_less_iff_less
tff(fact_2366_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% neg_less_iff_less
tff(fact_2367_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).

% neg_less_iff_less
tff(fact_2368_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_2369_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_2370_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_2371_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_2372_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_2373_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_2374_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_2375_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_2376_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_2377_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_2378_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_2379_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_2380_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_2381_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_2382_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_2383_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_2384_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_2385_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_2386_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_2387_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_2388_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_2389_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_2390_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_2391_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_2392_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_2393_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_2394_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_2395_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_2396_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_2397_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_2398_power__one,axiom,
    ! [N2: nat] : aa_nat_rat(power_power_rat(one_one_rat),N2) = one_one_rat ).

% power_one
tff(fact_2399_power__one,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N2) = one_one_nat ).

% power_one
tff(fact_2400_power__one,axiom,
    ! [N2: nat] : aa_nat_real(power_power_real(one_one_real),N2) = one_one_real ).

% power_one
tff(fact_2401_power__one,axiom,
    ! [N2: nat] : aa_nat_int(power_power_int(one_one_int),N2) = one_one_int ).

% power_one
tff(fact_2402_power__one,axiom,
    ! [N2: nat] : aa_nat_complex(power_power_complex(one_one_complex),N2) = one_one_complex ).

% power_one
tff(fact_2403_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_2404_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_2405_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_2406_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_2407_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_2408_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2409_power__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2410_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2411_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_2412_power__one__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),one_one_nat) = A ).

% power_one_right
tff(fact_2413_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_2414_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_2415_set__decode__inverse,axiom,
    ! [N2: nat] : aa_set_nat_nat(nat_set_encode,aa_nat_set_nat(nat_set_decode,N2)) = N2 ).

% set_decode_inverse
tff(fact_2416_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_less_eq_nonneg
tff(fact_2417_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_eq_nonneg
tff(fact_2418_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_less_eq_nonneg
tff(fact_2419_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_less_eq_nonneg
tff(fact_2420_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% less_eq_neg_nonpos
tff(fact_2421_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_eq_neg_nonpos
tff(fact_2422_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% less_eq_neg_nonpos
tff(fact_2423_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% less_eq_neg_nonpos
tff(fact_2424_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% neg_le_0_iff_le
tff(fact_2425_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_le_0_iff_le
tff(fact_2426_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% neg_le_0_iff_le
tff(fact_2427_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% neg_le_0_iff_le
tff(fact_2428_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% neg_0_le_iff_le
tff(fact_2429_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_le_iff_le
tff(fact_2430_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% neg_0_le_iff_le
tff(fact_2431_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).

% neg_0_le_iff_le
tff(fact_2432_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_0_iff_less
tff(fact_2433_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_0_iff_less
tff(fact_2434_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_0_iff_less
tff(fact_2435_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_0_iff_less
tff(fact_2436_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% neg_0_less_iff_less
tff(fact_2437_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% neg_0_less_iff_less
tff(fact_2438_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% neg_0_less_iff_less
tff(fact_2439_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% neg_0_less_iff_less
tff(fact_2440_neg__less__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).

% neg_less_pos
tff(fact_2441_neg__less__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% neg_less_pos
tff(fact_2442_neg__less__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% neg_less_pos
tff(fact_2443_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).

% neg_less_pos
tff(fact_2444_less__neg__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).

% less_neg_neg
tff(fact_2445_less__neg__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% less_neg_neg
tff(fact_2446_less__neg__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% less_neg_neg
tff(fact_2447_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).

% less_neg_neg
tff(fact_2448_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_2449_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_2450_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_2451_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_2452_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_2453_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_2454_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_2455_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_2456_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_2457_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_2458_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_2459_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_2460_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_2461_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_2462_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_2463_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_2464_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_2465_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_2466_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_2467_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_2468_power__inject__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_nat_real(power_power_real(A),M) = aa_nat_real(power_power_real(A),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2469_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2470_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2471_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N2) )
      <=> ( M = N2 ) ) ) ).

% power_inject_exp
tff(fact_2472_power__0__Suc,axiom,
    ! [N2: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N2)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2473_power__0__Suc,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N2)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2474_power__0__Suc,axiom,
    ! [N2: nat] : aa_nat_real(power_power_real(zero_zero_real),aa_nat_nat(suc,N2)) = zero_zero_real ).

% power_0_Suc
tff(fact_2475_power__0__Suc,axiom,
    ! [N2: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N2)) = zero_zero_int ).

% power_0_Suc
tff(fact_2476_power__0__Suc,axiom,
    ! [N2: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N2)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2477_mult__minus1,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z2) = aa_int_int(uminus_uminus_int,Z2) ).

% mult_minus1
tff(fact_2478_mult__minus1,axiom,
    ! [Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z2) = aa_real_real(uminus_uminus_real,Z2) ).

% mult_minus1
tff(fact_2479_mult__minus1,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z2) = aa_complex_complex(uminus1482373934393186551omplex,Z2) ).

% mult_minus1
tff(fact_2480_mult__minus1,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z2) = aa_rat_rat(uminus_uminus_rat,Z2) ).

% mult_minus1
tff(fact_2481_mult__minus1,axiom,
    ! [Z2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z2) ).

% mult_minus1
tff(fact_2482_mult__minus1__right,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z2),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z2) ).

% mult_minus1_right
tff(fact_2483_mult__minus1__right,axiom,
    ! [Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z2) ).

% mult_minus1_right
tff(fact_2484_mult__minus1__right,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(Z2),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z2) ).

% mult_minus1_right
tff(fact_2485_mult__minus1__right,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(Z2),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z2) ).

% mult_minus1_right
tff(fact_2486_mult__minus1__right,axiom,
    ! [Z2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z2) ).

% mult_minus1_right
tff(fact_2487_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_2488_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_2489_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_2490_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_2491_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_2492_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_2493_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_2494_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_2495_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_2496_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_2497_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_2498_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2499_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_2500_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_2501_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_2502_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_2503_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_2504_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_2505_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_2506_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_2507_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_2508_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_2509_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_2510_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_2511_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_2512_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_2513_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_2514_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_2515_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_2516_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_2517_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_2518_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_2519_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_2520_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_2521_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_2522_minus__one__mult__self,axiom,
    ! [N2: 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)),N2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2)) = one_one_int ).

% minus_one_mult_self
tff(fact_2523_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)) = one_one_real ).

% minus_one_mult_self
tff(fact_2524_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)) = one_one_complex ).

% minus_one_mult_self
tff(fact_2525_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)) = one_one_rat ).

% minus_one_mult_self
tff(fact_2526_minus__one__mult__self,axiom,
    ! [N2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_2527_left__minus__one__mult__self,axiom,
    ! [N2: 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)),N2)),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)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2528_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2529_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2530_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2531_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2532_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),X)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2533_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2534_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2535_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_2536_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( aa_nat_rat(power_power_rat(A),N2) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% power_eq_0_iff
tff(fact_2537_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% power_eq_0_iff
tff(fact_2538_power__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( aa_nat_real(power_power_real(A),N2) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% power_eq_0_iff
tff(fact_2539_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( aa_nat_int(power_power_int(A),N2) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% power_eq_0_iff
tff(fact_2540_power__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( aa_nat_complex(power_power_complex(A),N2) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% power_eq_0_iff
tff(fact_2541_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_2542_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_2543_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_2544_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_2545_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_2546_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),M)),aa_nat_real(power_power_real(B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2547_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2548_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2549_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N2)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2550_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),X)),aa_nat_real(power_power_real(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2551_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2552_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2553_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_2554_power__mono__iff,axiom,
    ! [A: real,B: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(B),N2)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2555_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)))
          <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2556_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2557_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(B),N2)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).

% power_mono_iff
tff(fact_2558_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_2559_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_2560_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_2561_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_2562_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_2563_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_2564_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_2565_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_2566_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_2567_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_2568_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_2569_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_2570_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_2571_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_2572_power__minus,axiom,
    ! [A: int,N2: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N2) = 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)),N2)),aa_nat_int(power_power_int(A),N2)) ).

% power_minus
tff(fact_2573_power__minus,axiom,
    ! [A: real,N2: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,A)),N2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)),aa_nat_real(power_power_real(A),N2)) ).

% power_minus
tff(fact_2574_power__minus,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)),aa_nat_complex(power_power_complex(A),N2)) ).

% power_minus
tff(fact_2575_power__minus,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),N2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),aa_nat_rat(power_power_rat(A),N2)) ).

% power_minus
tff(fact_2576_power__minus,axiom,
    ! [A: code_integer,N2: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)) ).

% power_minus
tff(fact_2577_power__not__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),N2) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2578_power__not__zero,axiom,
    ! [A: nat,N2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2579_power__not__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(power_power_real(A),N2) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2580_power__not__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),N2) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2581_power__not__zero,axiom,
    ! [A: complex,N2: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),N2) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2582_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),Y)) ) ).

% compl_le_swap2
tff(fact_2583_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X)))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y))) ) ).

% compl_le_swap1
tff(fact_2584_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X))) ) ).

% compl_mono
tff(fact_2585_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% le_minus_iff
tff(fact_2586_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_minus_iff
tff(fact_2587_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_minus_iff
tff(fact_2588_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% le_minus_iff
tff(fact_2589_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_le_iff
tff(fact_2590_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_le_iff
tff(fact_2591_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_le_iff
tff(fact_2592_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_le_iff
tff(fact_2593_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% le_imp_neg_le
tff(fact_2594_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% le_imp_neg_le
tff(fact_2595_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% le_imp_neg_le
tff(fact_2596_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% le_imp_neg_le
tff(fact_2597_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_2598_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_2599_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_2600_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% verit_negate_coefficient(2)
tff(fact_2601_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).

% less_minus_iff
tff(fact_2602_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(uminus_uminus_real,A))) ) ).

% less_minus_iff
tff(fact_2603_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A))) ) ).

% less_minus_iff
tff(fact_2604_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).

% less_minus_iff
tff(fact_2605_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).

% minus_less_iff
tff(fact_2606_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).

% minus_less_iff
tff(fact_2607_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A)) ) ).

% minus_less_iff
tff(fact_2608_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).

% minus_less_iff
tff(fact_2609_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_2610_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_2611_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_2612_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_2613_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_2614_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_2615_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_2616_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_2617_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_2618_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_2619_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M7)
     => ? [N: nat] :
        ! [X2: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X2,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X2)),N)) ) ) ).

% finite_maxlen
tff(fact_2620_finite__maxlen,axiom,
    ! [M7: set_list_bool] :
      ( finite7866827932528546246t_bool(M7)
     => ? [N: nat] :
        ! [X2: list_bool] :
          ( member_list_bool(X2,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X2)),N)) ) ) ).

% finite_maxlen
tff(fact_2621_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( finite8100373058378681591st_nat(M7)
     => ? [N: nat] :
        ! [X2: list_nat] :
          ( member_list_nat(X2,M7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X2)),N)) ) ) ).

% finite_maxlen
tff(fact_2622_length__induct,axiom,
    ! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs2)))
             => pp(aa_lis2822252094023462681T_bool(P,Ys)) )
         => pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
     => pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).

% length_induct
tff(fact_2623_length__induct,axiom,
    ! [P: fun_list_bool_bool,Xs: list_bool] :
      ( ! [Xs2: list_bool] :
          ( ! [Ys: list_bool] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys)),size_size_list_bool(Xs2)))
             => pp(aa_list_bool_bool(P,Ys)) )
         => pp(aa_list_bool_bool(P,Xs2)) )
     => pp(aa_list_bool_bool(P,Xs)) ) ).

% length_induct
tff(fact_2624_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys)),size_size_list_nat(Xs2)))
             => pp(aa_list_nat_bool(P,Ys)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_2625_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_2626_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_2627_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_2628_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_2629_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_2630_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_2631_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_2632_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_2633_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_2634_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_2635_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_2636_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_2637_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_2638_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_2639_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_2640_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_2641_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_2642_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_2643_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_2644_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_2645_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_2646_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_2647_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_2648_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_2649_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_2650_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_2651_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_2652_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_2653_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) ).

% minus_diff_minus
tff(fact_2654_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_2655_Ints__power,axiom,
    ! [A: real,N2: nat] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_nat_real(power_power_real(A),N2),ring_1_Ints_real) ) ).

% Ints_power
tff(fact_2656_Ints__power,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => pp(aa_set_int_bool(member_int(aa_nat_int(power_power_int(A),N2)),ring_1_Ints_int)) ) ).

% Ints_power
tff(fact_2657_Ints__power,axiom,
    ! [A: complex,N2: nat] :
      ( pp(aa_set_complex_bool(member_complex(A),ring_1_Ints_complex))
     => pp(aa_set_complex_bool(member_complex(aa_nat_complex(power_power_complex(A),N2)),ring_1_Ints_complex)) ) ).

% Ints_power
tff(fact_2658_minus__in__Ints__iff,axiom,
    ! [X: int] :
      ( pp(aa_set_int_bool(member_int(aa_int_int(uminus_uminus_int,X)),ring_1_Ints_int))
    <=> pp(aa_set_int_bool(member_int(X),ring_1_Ints_int)) ) ).

% minus_in_Ints_iff
tff(fact_2659_minus__in__Ints__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(uminus_uminus_real,X),ring_1_Ints_real)
    <=> member_real(X,ring_1_Ints_real) ) ).

% minus_in_Ints_iff
tff(fact_2660_minus__in__Ints__iff,axiom,
    ! [X: complex] :
      ( pp(aa_set_complex_bool(member_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),ring_1_Ints_complex))
    <=> pp(aa_set_complex_bool(member_complex(X),ring_1_Ints_complex)) ) ).

% minus_in_Ints_iff
tff(fact_2661_minus__in__Ints__iff,axiom,
    ! [X: rat] :
      ( member_rat(aa_rat_rat(uminus_uminus_rat,X),ring_1_Ints_rat)
    <=> member_rat(X,ring_1_Ints_rat) ) ).

% minus_in_Ints_iff
tff(fact_2662_minus__in__Ints__iff,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),ring_11222124179247155820nteger)
    <=> member_Code_integer(X,ring_11222124179247155820nteger) ) ).

% minus_in_Ints_iff
tff(fact_2663_Ints__minus,axiom,
    ! [A: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => pp(aa_set_int_bool(member_int(aa_int_int(uminus_uminus_int,A)),ring_1_Ints_int)) ) ).

% Ints_minus
tff(fact_2664_Ints__minus,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_real_real(uminus_uminus_real,A),ring_1_Ints_real) ) ).

% Ints_minus
tff(fact_2665_Ints__minus,axiom,
    ! [A: complex] :
      ( pp(aa_set_complex_bool(member_complex(A),ring_1_Ints_complex))
     => pp(aa_set_complex_bool(member_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),ring_1_Ints_complex)) ) ).

% Ints_minus
tff(fact_2666_Ints__minus,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(aa_rat_rat(uminus_uminus_rat,A),ring_1_Ints_rat) ) ).

% Ints_minus
tff(fact_2667_Ints__minus,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),ring_11222124179247155820nteger) ) ).

% Ints_minus
tff(fact_2668_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% subset_decode_imp_le
tff(fact_2669_power__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(B),N2))) ) ) ).

% power_mono
tff(fact_2670_power__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))) ) ) ).

% power_mono
tff(fact_2671_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))) ) ) ).

% power_mono
tff(fact_2672_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(B),N2))) ) ) ).

% power_mono
tff(fact_2673_zero__le__power,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N2))) ) ).

% zero_le_power
tff(fact_2674_zero__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))) ) ).

% zero_le_power
tff(fact_2675_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ).

% zero_le_power
tff(fact_2676_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N2))) ) ).

% zero_le_power
tff(fact_2677_zero__less__power,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),N2))) ) ).

% zero_less_power
tff(fact_2678_zero__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))) ) ).

% zero_less_power
tff(fact_2679_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ).

% zero_less_power
tff(fact_2680_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N2))) ) ).

% zero_less_power
tff(fact_2681_finite__set__decode,axiom,
    ! [N2: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(nat_set_decode,N2))) ).

% finite_set_decode
tff(fact_2682_one__le__power,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(A),N2))) ) ).

% one_le_power
tff(fact_2683_one__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N2))) ) ).

% one_le_power
tff(fact_2684_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ).

% one_le_power
tff(fact_2685_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N2))) ) ).

% one_le_power
tff(fact_2686_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N2: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N2)),aa_nat_complex(power_power_complex(Y),N2)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_2687_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N2: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),N2)),aa_nat_real(power_power_real(Y),N2)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_2688_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N2)),aa_nat_rat(power_power_rat(Y),N2)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2689_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N2)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2690_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N2)),aa_nat_int(power_power_int(Y),N2)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2691_le__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% le_minus_one_simps(2)
tff(fact_2692_le__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% le_minus_one_simps(2)
tff(fact_2693_le__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% le_minus_one_simps(2)
tff(fact_2694_le__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% le_minus_one_simps(2)
tff(fact_2695_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(4)
tff(fact_2696_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(4)
tff(fact_2697_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(4)
tff(fact_2698_le__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(4)
tff(fact_2699_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_2700_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_2701_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_2702_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2703_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_2704_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_2705_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_2706_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_2707_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2708_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_2709_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_2710_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_2711_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_2712_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_2713_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_2714_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_2715_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_2716_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_2717_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_2718_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_2719_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_2720_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_2721_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_2722_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_2723_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_2724_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_2725_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_2726_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_2727_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_2728_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_2729_less__minus__one__simps_I2_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).

% less_minus_one_simps(2)
tff(fact_2730_less__minus__one__simps_I2_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).

% less_minus_one_simps(2)
tff(fact_2731_less__minus__one__simps_I2_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat)) ).

% less_minus_one_simps(2)
tff(fact_2732_less__minus__one__simps_I2_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).

% less_minus_one_simps(2)
tff(fact_2733_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(4)
tff(fact_2734_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(4)
tff(fact_2735_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(4)
tff(fact_2736_less__minus__one__simps_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(4)
tff(fact_2737_power__Suc2,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N2)),A) ).

% power_Suc2
tff(fact_2738_power__Suc2,axiom,
    ! [A: real,N2: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),N2)),A) ).

% power_Suc2
tff(fact_2739_power__Suc2,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N2)),A) ).

% power_Suc2
tff(fact_2740_power__Suc2,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),A) ).

% power_Suc2
tff(fact_2741_power__Suc2,axiom,
    ! [A: int,N2: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N2)),A) ).

% power_Suc2
tff(fact_2742_power__Suc,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N2)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N2)) ).

% power_Suc
tff(fact_2743_power__Suc,axiom,
    ! [A: real,N2: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N2)) ).

% power_Suc
tff(fact_2744_power__Suc,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N2)) ).

% power_Suc
tff(fact_2745_power__Suc,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ).

% power_Suc
tff(fact_2746_power__Suc,axiom,
    ! [A: int,N2: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N2)) ).

% power_Suc
tff(fact_2747_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_2748_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_2749_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_2750_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_2751_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_2752_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2753_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_2754_power__0,axiom,
    ! [A: real] : aa_nat_real(power_power_real(A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2755_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2756_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2757_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_2758_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_2759_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_2760_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2761_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_2762_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_2763_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_2764_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_2765_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_2766_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_2767_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_2768_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_2769_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_2770_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_2771_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_2772_power__add,axiom,
    ! [A: complex,M: nat,N2: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N2)) ).

% power_add
tff(fact_2773_power__add,axiom,
    ! [A: real,M: nat,N2: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N2)) ).

% power_add
tff(fact_2774_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N2)) ).

% power_add
tff(fact_2775_power__add,axiom,
    ! [A: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ).

% power_add
tff(fact_2776_power__add,axiom,
    ! [A: int,M: nat,N2: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N2)) ).

% power_add
tff(fact_2777_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% nat_power_less_imp_less
tff(fact_2778_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),X)),size_s6755466524823107622T_VEBT(Xs))) ).

% count_le_length
tff(fact_2779_count__le__length,axiom,
    ! [Xs: list_bool,X: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_bool_nat(count_list_bool(Xs),X)),size_size_list_bool(Xs))) ).

% count_le_length
tff(fact_2780_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),X)),size_size_list_nat(Xs))) ).

% count_le_length
tff(fact_2781_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,zero_zero_nat,Ts2,S),X) = vEBT_Node(Info2,zero_zero_nat,Ts2,S) ).

% vebt_insert.simps(2)
tff(fact_2782_power__less__imp__less__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(B),N2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2783_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2784_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2785_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(B),N2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% power_less_imp_less_base
tff(fact_2786_power__le__one,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),one_one_real)) ) ) ).

% power_le_one
tff(fact_2787_power__le__one,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),one_one_rat)) ) ) ).

% power_le_one
tff(fact_2788_power__le__one,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),one_one_nat)) ) ) ).

% power_le_one
tff(fact_2789_power__le__one,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),one_one_int)) ) ) ).

% power_le_one
tff(fact_2790_power__inject__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),aa_nat_nat(suc,N2)) = aa_nat_real(power_power_real(B),aa_nat_nat(suc,N2)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2791_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N2)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2792_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2793_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N2)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N2)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_2794_power__le__imp__le__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N2))),aa_nat_real(power_power_real(B),aa_nat_nat(suc,N2))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2795_power__le__imp__le__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N2))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2796_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N2))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2797_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N2))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N2))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% power_le_imp_le_base
tff(fact_2798_power__less__power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N2)))) ) ).

% power_less_power_Suc
tff(fact_2799_power__less__power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N2)))) ) ).

% power_less_power_Suc
tff(fact_2800_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)))) ) ).

% power_less_power_Suc
tff(fact_2801_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N2)))) ) ).

% power_less_power_Suc
tff(fact_2802_power__gt1__lemma,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N2)))) ) ).

% power_gt1_lemma
tff(fact_2803_power__gt1__lemma,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N2)))) ) ).

% power_gt1_lemma
tff(fact_2804_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)))) ) ).

% power_gt1_lemma
tff(fact_2805_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N2)))) ) ).

% power_gt1_lemma
tff(fact_2806_le__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% le_minus_one_simps(1)
tff(fact_2807_le__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% le_minus_one_simps(1)
tff(fact_2808_le__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% le_minus_one_simps(1)
tff(fact_2809_le__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% le_minus_one_simps(1)
tff(fact_2810_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% le_minus_one_simps(3)
tff(fact_2811_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% le_minus_one_simps(3)
tff(fact_2812_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% le_minus_one_simps(3)
tff(fact_2813_le__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% le_minus_one_simps(3)
tff(fact_2814_less__minus__one__simps_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).

% less_minus_one_simps(1)
tff(fact_2815_less__minus__one__simps_I1_J,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).

% less_minus_one_simps(1)
tff(fact_2816_less__minus__one__simps_I1_J,axiom,
    pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat)) ).

% less_minus_one_simps(1)
tff(fact_2817_less__minus__one__simps_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).

% less_minus_one_simps(1)
tff(fact_2818_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).

% less_minus_one_simps(3)
tff(fact_2819_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).

% less_minus_one_simps(3)
tff(fact_2820_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% less_minus_one_simps(3)
tff(fact_2821_less__minus__one__simps_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% less_minus_one_simps(3)
tff(fact_2822_power__gt1,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,N2)))) ) ).

% power_gt1
tff(fact_2823_power__gt1,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2)))) ) ).

% power_gt1
tff(fact_2824_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2)))) ) ).

% power_gt1
tff(fact_2825_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N2)))) ) ).

% power_gt1
tff(fact_2826_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N2) = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(zero_zero_rat),N2) = zero_zero_rat ) ) ) ).

% power_0_left
tff(fact_2827_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N2) = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N2) = zero_zero_nat ) ) ) ).

% power_0_left
tff(fact_2828_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N2) = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_real(power_power_real(zero_zero_real),N2) = zero_zero_real ) ) ) ).

% power_0_left
tff(fact_2829_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N2) = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_int(power_power_int(zero_zero_int),N2) = zero_zero_int ) ) ) ).

% power_0_left
tff(fact_2830_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N2) = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(zero_zero_complex),N2) = zero_zero_complex ) ) ) ).

% power_0_left
tff(fact_2831_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% power_less_imp_less_exp
tff(fact_2832_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% power_less_imp_less_exp
tff(fact_2833_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% power_less_imp_less_exp
tff(fact_2834_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ) ).

% power_less_imp_less_exp
tff(fact_2835_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_2836_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_2837_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5))) ) ) ).

% power_strict_increasing
tff(fact_2838_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(A),N5))) ) ) ).

% power_strict_increasing
tff(fact_2839_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(A),N5))) ) ) ).

% power_increasing
tff(fact_2840_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),N5))) ) ) ).

% power_increasing
tff(fact_2841_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5))) ) ) ).

% power_increasing
tff(fact_2842_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(A),N5))) ) ) ).

% power_increasing
tff(fact_2843_zero__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N2) = zero_zero_rat ) ) ).

% zero_power
tff(fact_2844_zero__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N2) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2845_zero__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_real(power_power_real(zero_zero_real),N2) = zero_zero_real ) ) ).

% zero_power
tff(fact_2846_zero__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_int(power_power_int(zero_zero_int),N2) = zero_zero_int ) ) ).

% zero_power
tff(fact_2847_zero__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),N2) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2848_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2849_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2850_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2851_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2852_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2853_length__pos__if__in__set,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_bool(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2854_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_2855_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),K))) ) ).

% power_gt_expt
tff(fact_2856_nat__one__le__power,axiom,
    ! [I: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N2))) ) ).

% nat_one_le_power
tff(fact_2857_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_2858_power__Suc__less,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(A),N2))),aa_nat_real(power_power_real(A),N2))) ) ) ).

% power_Suc_less
tff(fact_2859_power__Suc__less,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N2))),aa_nat_rat(power_power_rat(A),N2))) ) ) ).

% power_Suc_less
tff(fact_2860_power__Suc__less,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ) ).

% power_Suc_less
tff(fact_2861_power__Suc__less,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N2))),aa_nat_int(power_power_int(A),N2))) ) ) ).

% power_Suc_less
tff(fact_2862_power__Suc__le__self,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N2))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2863_power__Suc__le__self,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2864_power__Suc__le__self,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2865_power__Suc__le__self,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N2))),A)) ) ) ).

% power_Suc_le_self
tff(fact_2866_power__Suc__less__one,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_nat_nat(suc,N2))),one_one_real)) ) ) ).

% power_Suc_less_one
tff(fact_2867_power__Suc__less__one,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N2))),one_one_rat)) ) ) ).

% power_Suc_less_one
tff(fact_2868_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N2))),one_one_nat)) ) ) ).

% power_Suc_less_one
tff(fact_2869_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N2))),one_one_int)) ) ) ).

% power_Suc_less_one
tff(fact_2870_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N5)),aa_nat_real(power_power_real(A),N2))) ) ) ) ).

% power_strict_decreasing
tff(fact_2871_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N5)),aa_nat_rat(power_power_rat(A),N2))) ) ) ) ).

% power_strict_decreasing
tff(fact_2872_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ) ) ).

% power_strict_decreasing
tff(fact_2873_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N5)),aa_nat_int(power_power_int(A),N2))) ) ) ) ).

% power_strict_decreasing
tff(fact_2874_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N5)),aa_nat_real(power_power_real(A),N2))) ) ) ) ).

% power_decreasing
tff(fact_2875_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N5)),aa_nat_rat(power_power_rat(A),N2))) ) ) ) ).

% power_decreasing
tff(fact_2876_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ) ) ).

% power_decreasing
tff(fact_2877_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N5)),aa_nat_int(power_power_int(A),N2))) ) ) ) ).

% power_decreasing
tff(fact_2878_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% power_le_imp_le_exp
tff(fact_2879_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% power_le_imp_le_exp
tff(fact_2880_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% power_le_imp_le_exp
tff(fact_2881_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N2)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% power_le_imp_le_exp
tff(fact_2882_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( ( aa_nat_real(power_power_real(A),N2) = aa_nat_real(power_power_real(B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2883_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( ( aa_nat_rat(power_power_rat(A),N2) = aa_nat_rat(power_power_rat(B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2884_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2885_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( ( aa_nat_int(power_power_int(A),N2) = aa_nat_int(power_power_int(B),N2) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2886_power__eq__imp__eq__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( aa_nat_real(power_power_real(A),N2) = aa_nat_real(power_power_real(B),N2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2887_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),N2) = aa_nat_rat(power_power_rat(B),N2) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2888_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2889_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),N2) = aa_nat_int(power_power_int(B),N2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2890_self__le__power,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(power_power_real(A),N2))) ) ) ).

% self_le_power
tff(fact_2891_self__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N2))) ) ) ).

% self_le_power
tff(fact_2892_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ) ).

% self_le_power
tff(fact_2893_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N2))) ) ) ).

% self_le_power
tff(fact_2894_one__less__power,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(power_power_real(A),N2))) ) ) ).

% one_less_power
tff(fact_2895_one__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N2))) ) ) ).

% one_less_power
tff(fact_2896_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ) ).

% one_less_power
tff(fact_2897_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N2))) ) ) ).

% one_less_power
tff(fact_2898_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X) = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ).

% vebt_insert.simps(3)
tff(fact_2899_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: bool,B: bool] :
      ( ( ( X = zero_zero_nat )
       => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(fTrue,B) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,fTrue) ) )
          & ( ( X != one_one_nat )
           => ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,B) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_2900_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2901_power__strict__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(B),N2))) ) ) ) ).

% power_strict_mono
tff(fact_2902_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))) ) ) ) ).

% power_strict_mono
tff(fact_2903_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2))) ) ) ) ).

% power_strict_mono
tff(fact_2904_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(B),N2))) ) ) ) ).

% power_strict_mono
tff(fact_2905_power__eq__if,axiom,
    ! [M: nat,P3: complex] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(power_power_complex(P3),M) = aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2906_power__eq__if,axiom,
    ! [M: nat,P3: real] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(power_power_real(P3),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_nat_real(power_power_real(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2907_power__eq__if,axiom,
    ! [M: nat,P3: rat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(power_power_rat(P3),M) = aa_rat_rat(times_times_rat(P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2908_power__eq__if,axiom,
    ! [M: nat,P3: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2909_power__eq__if,axiom,
    ! [M: nat,P3: int] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(power_power_int(P3),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% power_eq_if
tff(fact_2910_power__minus__mult,axiom,
    ! [N2: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N2) ) ) ).

% power_minus_mult
tff(fact_2911_power__minus__mult,axiom,
    ! [N2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),A) = aa_nat_real(power_power_real(A),N2) ) ) ).

% power_minus_mult
tff(fact_2912_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N2) ) ) ).

% power_minus_mult
tff(fact_2913_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( 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,N2),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2) ) ) ).

% power_minus_mult
tff(fact_2914_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( 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,N2),one_one_nat))),A) = aa_nat_int(power_power_int(A),N2) ) ) ).

% power_minus_mult
tff(fact_2915_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( 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,N2),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2916_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K)) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2917_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2918_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2919_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2920_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_2921_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_2922_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ( aa_nat_real(power_power_real(R3),N2) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2923_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
            & ( aa_nat_real(power_power_real(X4),N2) = A )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4))
                  & ( aa_nat_real(power_power_real(Y4),N2) = A ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2924__C5_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I4 )
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
          & ! [X2: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),na) = I4 )
                & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),na))) )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),X2))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),ma)) ) ) ) ) ) ).

% "5.hyps"(9)
tff(fact_2925_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,X)),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2926_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2927_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2928_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2929_power__minus_H,axiom,
    ! [X: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,one_one_int),X))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),N2) = 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)),N2)),aa_nat_int(power_power_int(X),N2)) ) ) ).

% power_minus'
tff(fact_2930_power__minus_H,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(nO_MATCH_real_real,one_one_real),X))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,X)),N2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)),aa_nat_real(power_power_real(X),N2)) ) ) ).

% power_minus'
tff(fact_2931_power__minus_H,axiom,
    ! [X: complex,N2: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(nO_MAT8947977539597988553omplex,one_one_complex),X))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),N2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)),aa_nat_complex(power_power_complex(X),N2)) ) ) ).

% power_minus'
tff(fact_2932_power__minus_H,axiom,
    ! [X: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(nO_MATCH_rat_rat,one_one_rat),X))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),N2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),aa_nat_rat(power_power_rat(X),N2)) ) ) ).

% power_minus'
tff(fact_2933_power__minus_H,axiom,
    ! [X: code_integer,N2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(nO_MAT8252062027627875367nteger,one_one_Code_integer),X))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N2)) ) ) ).

% power_minus'
tff(fact_2934_prod__decode__triangle__add,axiom,
    ! [K: nat,M: nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M)) = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(K),M) ).

% prod_decode_triangle_add
tff(fact_2935_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,N2: nat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),set_complex2(Xs)))
         => pp(aa_complex_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s3451745648224563538omplex(Xs)))
       => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),N2))) ) ) ).

% inthall
tff(fact_2936_inthall,axiom,
    ! [Xs: list_real,P: fun_real_bool,N2: nat] :
      ( ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => pp(aa_real_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_real(Xs)))
       => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N2))) ) ) ).

% inthall
tff(fact_2937_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,N2: nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,set_set_nat2(Xs))
         => pp(aa_set_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s3254054031482475050et_nat(Xs)))
       => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),N2))) ) ) ).

% inthall
tff(fact_2938_inthall,axiom,
    ! [Xs: list_int,P: fun_int_bool,N2: nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => pp(aa_int_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_int(Xs)))
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N2))) ) ) ).

% inthall
tff(fact_2939_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N2: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s6755466524823107622T_VEBT(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N2))) ) ) ).

% inthall
tff(fact_2940_inthall,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,N2: nat] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => pp(aa_bool_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_bool(Xs)))
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N2))) ) ) ).

% inthall
tff(fact_2941_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,N2: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X4)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N2))) ) ) ).

% inthall
tff(fact_2942__C5_Ohyps_C_I2_J,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "5.hyps"(2)
tff(fact_2943_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( numera1916890842035813515d_enat(M) = numera1916890842035813515d_enat(N2) )
    <=> ( M = N2 ) ) ).

% numeral_eq_iff
tff(fact_2944_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N2) )
    <=> ( M = N2 ) ) ).

% numeral_eq_iff
tff(fact_2945_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,N2) )
    <=> ( M = N2 ) ) ).

% numeral_eq_iff
tff(fact_2946_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N2) )
    <=> ( M = N2 ) ) ).

% numeral_eq_iff
tff(fact_2947_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N2) )
    <=> ( M = N2 ) ) ).

% numeral_eq_iff
tff(fact_2948_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% semiring_norm(78)
tff(fact_2949_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_2950_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),one)) ).

% semiring_norm(75)
tff(fact_2951_negative__zless,axiom,
    ! [N2: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zless
tff(fact_2952_negative__eq__positive,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_2953__C5_Ohyps_C_I8_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).

% "5.hyps"(8)
tff(fact_2954_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X = A ) ) ).

% real_add_minus_iff
tff(fact_2955_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_2956_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_2957_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = aa_nat_rat(semiri681578069525770553at_rat,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_2958_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_2959_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( M = N2 ) ) ).

% of_nat_eq_iff
tff(fact_2960_prod__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode,X) = aa_nat7603221163087113075at_nat(nat_prod_decode,Y) )
    <=> ( X = Y ) ) ).

% prod_decode_eq
tff(fact_2961_dp,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% dp
tff(fact_2962_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ).

% high_bound_aux
tff(fact_2963__C5_Ohyps_C_I5_J,axiom,
    ! [I4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I4)),X_13))
      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I4)) ) ) ).

% "5.hyps"(5)
tff(fact_2964_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) ) ).

% member_bound
tff(fact_2965_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2)) ) ).

% numeral_le_iff
tff(fact_2966_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2)) ) ).

% numeral_le_iff
tff(fact_2967_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2)) ) ).

% numeral_le_iff
tff(fact_2968_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2)) ) ).

% numeral_le_iff
tff(fact_2969_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2)) ) ).

% numeral_le_iff
tff(fact_2970_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% numeral_less_iff
tff(fact_2971_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% numeral_less_iff
tff(fact_2972_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% numeral_less_iff
tff(fact_2973_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% numeral_less_iff
tff(fact_2974_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% numeral_less_iff
tff(fact_2975_high__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
     => ( 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))),N2))),X)),N2) = Y ) ) ).

% high_inv
tff(fact_2976_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N2)) ).

% numeral_times_numeral
tff(fact_2977_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N2)) = numera1916890842035813515d_enat(aa_num_num(times_times_num(M),N2)) ).

% numeral_times_numeral
tff(fact_2978_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N2)) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),N2)) ).

% numeral_times_numeral
tff(fact_2979_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N2)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N2)) ).

% numeral_times_numeral
tff(fact_2980_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),N2)) ).

% numeral_times_numeral
tff(fact_2981_numeral__times__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N2)) ).

% numeral_times_numeral
tff(fact_2982_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),Z2)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_2983_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(W2)),Z2)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(times_times_num(V),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_2984_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),Z2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_2985_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_2986_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W2)),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(V),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_2987_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W2)),Z2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W2))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_2988_low__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
     => ( 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))),N2))),X)),N2) = X ) ) ).

% low_inv
tff(fact_2989_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W2)),Z2)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W2))),Z2) ).

% add_numeral_left
tff(fact_2990_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2)),Z2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(aa_num_num(plus_plus_num(V),W2))),Z2) ).

% add_numeral_left
tff(fact_2991_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W2)),Z2)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W2))),Z2) ).

% add_numeral_left
tff(fact_2992_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W2)),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W2))),Z2) ).

% add_numeral_left
tff(fact_2993_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W2)),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W2))),Z2) ).

% add_numeral_left
tff(fact_2994_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W2)),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W2))),Z2) ).

% add_numeral_left
tff(fact_2995_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N2)) ).

% numeral_plus_numeral
tff(fact_2996_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N2)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(M),N2)) ).

% numeral_plus_numeral
tff(fact_2997_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N2)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),N2)) ).

% numeral_plus_numeral
tff(fact_2998_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N2)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N2)) ).

% numeral_plus_numeral
tff(fact_2999_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),N2)) ).

% numeral_plus_numeral
tff(fact_3000_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N2)) ).

% numeral_plus_numeral
tff(fact_3001_semiring__norm_I76_J,axiom,
    ! [N2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit0,N2))) ).

% semiring_norm(76)
tff(fact_3002_num__double,axiom,
    ! [N2: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),N2) = aa_num_num(bit0,N2) ).

% num_double
tff(fact_3003_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) )
    <=> ( M = N2 ) ) ).

% neg_numeral_eq_iff
tff(fact_3004_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)) )
    <=> ( M = N2 ) ) ).

% neg_numeral_eq_iff
tff(fact_3005_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2)) )
    <=> ( M = N2 ) ) ).

% neg_numeral_eq_iff
tff(fact_3006_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)) )
    <=> ( M = N2 ) ) ).

% neg_numeral_eq_iff
tff(fact_3007_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)) )
    <=> ( M = N2 ) ) ).

% neg_numeral_eq_iff
tff(fact_3008_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_3009_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2)) ) ) ).

% set_n_deg_not_0
tff(fact_3010_misiz,axiom,
    ! [T: vEBT_VEBT,N2: nat,M: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) ) ).

% misiz
tff(fact_3011_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3012_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_3013_helpyd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_succ(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) ) ).

% helpyd
tff(fact_3014_helpypredd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_pred(T,X) = some_nat(Y) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) ) ).

% helpypredd
tff(fact_3015_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
         => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
           => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_3016_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
       => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_3017_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
         => ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y))
           => ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_3018_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_3019__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_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))) ).

% \<open>ma \<le> 2 ^ deg\<close>
tff(fact_3020_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary) ) ) ) ).

% del_single_cont
tff(fact_3021_xdegrel,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).

% xdegrel
tff(fact_3022_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_3023_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_3024_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_3025_prod__decode__inverse,axiom,
    ! [N2: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) = N2 ).

% prod_decode_inverse
tff(fact_3026_prod__encode__inverse,axiom,
    ! [X: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(nat_prod_decode,aa_Pro4243173770044397973at_nat(nat_prod_encode,X)) = X ).

% prod_encode_inverse
tff(fact_3027_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),M)) ) ).

% neg_numeral_le_iff
tff(fact_3028_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),M)) ) ).

% neg_numeral_le_iff
tff(fact_3029_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),M)) ) ).

% neg_numeral_le_iff
tff(fact_3030_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),M)) ) ).

% neg_numeral_le_iff
tff(fact_3031_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_3032_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),numera1916890842035813515d_enat(V)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),numera1916890842035813515d_enat(V))),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(B),numera1916890842035813515d_enat(V))) ).

% distrib_right_numeral
tff(fact_3033_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_3034_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_3035_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_3036_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_3037_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_3038_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(B),C)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),B)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V)),C)) ).

% distrib_left_numeral
tff(fact_3039_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_3040_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_3041_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_3042_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_3043_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N2),M)) ) ).

% neg_numeral_less_iff
tff(fact_3044_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N2),M)) ) ).

% neg_numeral_less_iff
tff(fact_3045_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N2),M)) ) ).

% neg_numeral_less_iff
tff(fact_3046_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N2),M)) ) ).

% neg_numeral_less_iff
tff(fact_3047_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,N2) )
    <=> ( one = N2 ) ) ).

% one_eq_numeral_iff
tff(fact_3048_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_on7984719198319812577d_enat = numera1916890842035813515d_enat(N2) )
    <=> ( one = N2 ) ) ).

% one_eq_numeral_iff
tff(fact_3049_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(N2) )
    <=> ( one = N2 ) ) ).

% one_eq_numeral_iff
tff(fact_3050_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,N2) )
    <=> ( one = N2 ) ) ).

% one_eq_numeral_iff
tff(fact_3051_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N2) )
    <=> ( one = N2 ) ) ).

% one_eq_numeral_iff
tff(fact_3052_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N2) )
    <=> ( one = N2 ) ) ).

% one_eq_numeral_iff
tff(fact_3053_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_num_rat(numeral_numeral_rat,N2) = one_one_rat )
    <=> ( N2 = one ) ) ).

% numeral_eq_one_iff
tff(fact_3054_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat(N2) = one_on7984719198319812577d_enat )
    <=> ( N2 = one ) ) ).

% numeral_eq_one_iff
tff(fact_3055_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex(N2) = one_one_complex )
    <=> ( N2 = one ) ) ).

% numeral_eq_one_iff
tff(fact_3056_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_num_real(numeral_numeral_real,N2) = one_one_real )
    <=> ( N2 = one ) ) ).

% numeral_eq_one_iff
tff(fact_3057_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N2) = one_one_nat )
    <=> ( N2 = one ) ) ).

% numeral_eq_one_iff
tff(fact_3058_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_num_int(numeral_numeral_int,N2) = one_one_int )
    <=> ( N2 = one ) ) ).

% numeral_eq_one_iff
tff(fact_3059_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_3060_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_3061_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_3062_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_3063_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3064_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_3065_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3066_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_3067_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_3068_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_3069_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2))) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_3070_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_3071_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))) = numera6620942414471956472nteger(aa_num_num(times_times_num(M),N2)) ).

% mult_neg_numeral_simps(1)
tff(fact_3072_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_3073_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N2)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_3074_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N2)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_3075_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_3076_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(2)
tff(fact_3077_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_3078_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_3079_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_3080_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_3081_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(M),N2))) ).

% mult_neg_numeral_simps(3)
tff(fact_3082_semiring__norm_I168_J,axiom,
    ! [V: num,W2: 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,W2))),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(plus_plus_num(V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_3083_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_3084_semiring__norm_I168_J,axiom,
    ! [V: num,W2: 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(W2))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_3085_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W2)))),Y) ).

% semiring_norm(168)
tff(fact_3086_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W2)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W2))),Y) ).

% semiring_norm(168)
tff(fact_3087_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))) ).

% add_neg_numeral_simps(3)
tff(fact_3088_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,N2))) ).

% add_neg_numeral_simps(3)
tff(fact_3089_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N2))) ).

% add_neg_numeral_simps(3)
tff(fact_3090_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,N2))) ).

% add_neg_numeral_simps(3)
tff(fact_3091_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N2))) ).

% add_neg_numeral_simps(3)
tff(fact_3092_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N2)) ).

% diff_numeral_simps(2)
tff(fact_3093_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N2)) ).

% diff_numeral_simps(2)
tff(fact_3094_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),N2)) ).

% diff_numeral_simps(2)
tff(fact_3095_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N2)) ).

% diff_numeral_simps(2)
tff(fact_3096_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),N2)) ).

% diff_numeral_simps(2)
tff(fact_3097_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),N2))) ).

% diff_numeral_simps(3)
tff(fact_3098_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N2)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),N2))) ).

% diff_numeral_simps(3)
tff(fact_3099_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N2)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),N2))) ).

% diff_numeral_simps(3)
tff(fact_3100_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),N2))) ).

% diff_numeral_simps(3)
tff(fact_3101_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),N2))) ).

% diff_numeral_simps(3)
tff(fact_3102_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_3103_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_3104_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_3105_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_3106_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_3107_Suc__numeral,axiom,
    ! [N2: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N2),one)) ).

% Suc_numeral
tff(fact_3108_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_3109_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_3110_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_3111_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_3112_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_3113_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3114_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3115_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3116_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3117_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
tff(fact_3118_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3119_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3120_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3121_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3122_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3123_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_iff
tff(fact_3124_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_iff
tff(fact_3125_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_iff
tff(fact_3126_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_iff
tff(fact_3127_of__nat__numeral,axiom,
    ! [N2: num] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_num_nat(numeral_numeral_nat,N2)) = numera1916890842035813515d_enat(N2) ).

% of_nat_numeral
tff(fact_3128_of__nat__numeral,axiom,
    ! [N2: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N2)) = numera6690914467698888265omplex(N2) ).

% of_nat_numeral
tff(fact_3129_of__nat__numeral,axiom,
    ! [N2: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_real(numeral_numeral_real,N2) ).

% of_nat_numeral
tff(fact_3130_of__nat__numeral,axiom,
    ! [N2: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_rat(numeral_numeral_rat,N2) ).

% of_nat_numeral
tff(fact_3131_of__nat__numeral,axiom,
    ! [N2: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,N2) ).

% of_nat_numeral
tff(fact_3132_of__nat__numeral,axiom,
    ! [N2: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_int(numeral_numeral_int,N2) ).

% of_nat_numeral
tff(fact_3133_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% of_nat_le_iff
tff(fact_3134_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% of_nat_le_iff
tff(fact_3135_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% of_nat_le_iff
tff(fact_3136_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% of_nat_le_iff
tff(fact_3137_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N2)) ).

% of_nat_add
tff(fact_3138_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% of_nat_add
tff(fact_3139_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_add
tff(fact_3140_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_add
tff(fact_3141_of__nat__add,axiom,
    ! [M: nat,N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_add
tff(fact_3142_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N2)) ).

% of_nat_mult
tff(fact_3143_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% of_nat_mult
tff(fact_3144_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% of_nat_mult
tff(fact_3145_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ).

% of_nat_mult
tff(fact_3146_of__nat__mult,axiom,
    ! [M: nat,N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% of_nat_mult
tff(fact_3147_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,N2) = one_one_complex )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3148_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N2) = one_one_real )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3149_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N2) = one_one_rat )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3150_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N2) = one_one_nat )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3151_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N2) = one_one_int )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3152_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3153_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3154_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3155_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3156_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3157_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_3158_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_3159_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3160_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3161_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3162_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N2)),one_on7984719198319812577d_enat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),one)) ) ).

% numeral_le_one_iff
tff(fact_3163_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N2)),one_one_real))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),one)) ) ).

% numeral_le_one_iff
tff(fact_3164_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),one_one_rat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),one)) ) ).

% numeral_le_one_iff
tff(fact_3165_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),one)) ) ).

% numeral_le_one_iff
tff(fact_3166_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),one_one_int))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N2),one)) ) ).

% numeral_le_one_iff
tff(fact_3167_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N2)) ) ).

% one_less_numeral_iff
tff(fact_3168_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N2)) ) ).

% one_less_numeral_iff
tff(fact_3169_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N2)) ) ).

% one_less_numeral_iff
tff(fact_3170_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N2)) ) ).

% one_less_numeral_iff
tff(fact_3171_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N2)) ) ).

% one_less_numeral_iff
tff(fact_3172_numeral__plus__one,axiom,
    ! [N2: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N2)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(N2),one)) ).

% numeral_plus_one
tff(fact_3173_numeral__plus__one,axiom,
    ! [N2: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N2)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(N2),one)) ).

% numeral_plus_one
tff(fact_3174_numeral__plus__one,axiom,
    ! [N2: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N2)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(N2),one)) ).

% numeral_plus_one
tff(fact_3175_numeral__plus__one,axiom,
    ! [N2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N2)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(N2),one)) ).

% numeral_plus_one
tff(fact_3176_numeral__plus__one,axiom,
    ! [N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(N2),one)) ).

% numeral_plus_one
tff(fact_3177_numeral__plus__one,axiom,
    ! [N2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N2)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(N2),one)) ).

% numeral_plus_one
tff(fact_3178_one__plus__numeral,axiom,
    ! [N2: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,N2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),N2)) ).

% one_plus_numeral
tff(fact_3179_one__plus__numeral,axiom,
    ! [N2: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N2)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(one),N2)) ).

% one_plus_numeral
tff(fact_3180_one__plus__numeral,axiom,
    ! [N2: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N2)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),N2)) ).

% one_plus_numeral
tff(fact_3181_one__plus__numeral,axiom,
    ! [N2: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,N2)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),N2)) ).

% one_plus_numeral
tff(fact_3182_one__plus__numeral,axiom,
    ! [N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),N2)) ).

% one_plus_numeral
tff(fact_3183_one__plus__numeral,axiom,
    ! [N2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N2)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N2)) ).

% one_plus_numeral
tff(fact_3184_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) )
    <=> ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3185_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)) )
    <=> ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3186_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2)) )
    <=> ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3187_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)) )
    <=> ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3188_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)) )
    <=> ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3189_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3190_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3191_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3192_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3193_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3194_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3195_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3196_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3197_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3198_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_3199_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_3200_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_3201_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_3202_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_3203_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3204_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3205_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3206_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3207_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3208_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3209_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3210_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3211_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_3212_one__add__one,axiom,
    aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_3213_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_3214_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_3215_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_3216_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_3217_diff__numeral__special_I3_J,axiom,
    ! [N2: 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,N2))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),N2)) ).

% diff_numeral_special(3)
tff(fact_3218_diff__numeral__special_I3_J,axiom,
    ! [N2: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),N2)) ).

% diff_numeral_special(3)
tff(fact_3219_diff__numeral__special_I3_J,axiom,
    ! [N2: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),N2)) ).

% diff_numeral_special(3)
tff(fact_3220_diff__numeral__special_I3_J,axiom,
    ! [N2: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),N2)) ).

% diff_numeral_special(3)
tff(fact_3221_diff__numeral__special_I3_J,axiom,
    ! [N2: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),N2)) ).

% diff_numeral_special(3)
tff(fact_3222_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_3223_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_3224_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_3225_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_3226_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_3227_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_3228_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_3229_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_3230_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_3231_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_3232_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% of_nat_0_less_iff
tff(fact_3233_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% of_nat_0_less_iff
tff(fact_3234_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% of_nat_0_less_iff
tff(fact_3235_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% of_nat_0_less_iff
tff(fact_3236_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_dec_simps(1)
tff(fact_3237_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_3238_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_3239_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_3240_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K))) ).

% dbl_dec_simps(1)
tff(fact_3241_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_inc_simps(1)
tff(fact_3242_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_3243_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_3244_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_3245_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K))) ).

% dbl_inc_simps(1)
tff(fact_3246_add__2__eq__Suc_H,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N2)) ).

% add_2_eq_Suc'
tff(fact_3247_add__2__eq__Suc,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = aa_nat_nat(suc,aa_nat_nat(suc,N2)) ).

% add_2_eq_Suc
tff(fact_3248_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_3249_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3250_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3251_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3252_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3253_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3254_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3255_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3256_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3257_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3258_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3259_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3260_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2))) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3261_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3262_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3263_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3264_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),X)) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3265_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3266_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3267_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3268_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3269_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_3270_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_3271_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_3272_zero__less__power2,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_3273_zero__less__power2,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_3274_zero__less__power2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_3275_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_3276_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_3277_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_3278_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_3279_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_3280_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_3281_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_3282_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_3283_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_3284_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_3285_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_3286_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_3287_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_3288_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_3289_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_3290_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3291_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3292_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3293_power__minus1__even,axiom,
    ! [N2: 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))),N2)) = one_one_int ).

% power_minus1_even
tff(fact_3294_power__minus1__even,axiom,
    ! [N2: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = one_one_real ).

% power_minus1_even
tff(fact_3295_power__minus1__even,axiom,
    ! [N2: 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))),N2)) = one_one_complex ).

% power_minus1_even
tff(fact_3296_power__minus1__even,axiom,
    ! [N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = one_one_rat ).

% power_minus1_even
tff(fact_3297_power__minus1__even,axiom,
    ! [N2: 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))),N2)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_3298_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3299_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3300_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3301_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3302_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3303_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3304_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3305_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3306_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N2)),aa_nat_real(semiri5074537144036343181t_real,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3307_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N2)),aa_nat_rat(semiri681578069525770553at_rat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3308_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3309_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N2)),aa_nat_int(semiri1314217659103216013at_int,X)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)),X)) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3310_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3311_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3312_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3313_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N2))) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3314_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ? [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),N)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_3315_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% realpow_square_minus_le
tff(fact_3316_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_less_0_iff
tff(fact_3317_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_3318_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).

% real_add_le_0_iff
tff(fact_3319_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).

% real_0_le_add_iff
tff(fact_3320_of__nat__less__two__power,axiom,
    ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N2))) ).

% of_nat_less_two_power
tff(fact_3321_of__nat__less__two__power,axiom,
    ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N2))) ).

% of_nat_less_two_power
tff(fact_3322_of__nat__less__two__power,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ).

% of_nat_less_two_power
tff(fact_3323_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ? [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(power_power_real(X),N))) ) ).

% real_arch_pow
tff(fact_3324_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% power2_nat_le_imp_le
tff(fact_3325_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% power2_nat_le_eq_le
tff(fact_3326_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_3327_less__exp,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% less_exp
tff(fact_3328_Bernoulli__inequality,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N2))) ) ).

% Bernoulli_inequality
tff(fact_3329_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_3330_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ! [Y4: real] :
        ? [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))) ) ).

% reals_Archimedean3
tff(fact_3331_Suc__double__not__eq__double,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) ).

% Suc_double_not_eq_double
tff(fact_3332_double__not__eq__Suc__double,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% double_not_eq_Suc_double
tff(fact_3333_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N2)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N2)),aa_num_rat(numeral_numeral_rat,N2)) ).

% numeral_Bit0
tff(fact_3334_numeral__Bit0,axiom,
    ! [N2: num] : numera1916890842035813515d_enat(aa_num_num(bit0,N2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N2)),numera1916890842035813515d_enat(N2)) ).

% numeral_Bit0
tff(fact_3335_numeral__Bit0,axiom,
    ! [N2: num] : numera6690914467698888265omplex(aa_num_num(bit0,N2)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N2)),numera6690914467698888265omplex(N2)) ).

% numeral_Bit0
tff(fact_3336_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N2)),aa_num_real(numeral_numeral_real,N2)) ).

% numeral_Bit0
tff(fact_3337_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,N2)) ).

% numeral_Bit0
tff(fact_3338_numeral__Bit0,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,N2)) ).

% numeral_Bit0
tff(fact_3339_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_3340_mult__numeral__1__right,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),numera1916890842035813515d_enat(one)) = A ).

% mult_numeral_1_right
tff(fact_3341_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_3342_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_3343_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_3344_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_3345_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_3346_mult__numeral__1,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(one)),A) = A ).

% mult_numeral_1
tff(fact_3347_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_3348_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_3349_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_3350_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_3351_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_3352_numeral__One,axiom,
    numera1916890842035813515d_enat(one) = one_on7984719198319812577d_enat ).

% numeral_One
tff(fact_3353_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_3354_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_3355_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_3356_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_3357_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_3358_add__One__commute,axiom,
    ! [N2: num] : aa_num_num(plus_plus_num(one),N2) = aa_num_num(plus_plus_num(N2),one) ).

% add_One_commute
tff(fact_3359_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),one))
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_3360_pos2,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% pos2
tff(fact_3361_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_3362_mult__2,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z2) = aa_rat_rat(plus_plus_rat(Z2),Z2) ).

% mult_2
tff(fact_3363_mult__2,axiom,
    ! [Z2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),Z2) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z2),Z2) ).

% mult_2
tff(fact_3364_mult__2,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z2) = aa_complex_complex(plus_plus_complex(Z2),Z2) ).

% mult_2
tff(fact_3365_mult__2,axiom,
    ! [Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),Z2) ).

% mult_2
tff(fact_3366_mult__2,axiom,
    ! [Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),Z2) ).

% mult_2
tff(fact_3367_mult__2,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z2) ).

% mult_2
tff(fact_3368_mult__2__right,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(Z2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z2),Z2) ).

% mult_2_right
tff(fact_3369_mult__2__right,axiom,
    ! [Z2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Z2),numera1916890842035813515d_enat(aa_num_num(bit0,one))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Z2),Z2) ).

% mult_2_right
tff(fact_3370_mult__2__right,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(Z2),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z2),Z2) ).

% mult_2_right
tff(fact_3371_mult__2__right,axiom,
    ! [Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),Z2) ).

% mult_2_right
tff(fact_3372_mult__2__right,axiom,
    ! [Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),Z2) ).

% mult_2_right
tff(fact_3373_mult__2__right,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z2) ).

% mult_2_right
tff(fact_3374_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3375_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3376_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_3377_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_3378_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_3379_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_3380_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_3381_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_3382_zero__power2,axiom,
    aa_nat_real(power_power_real(zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_3383_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_3384_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_3385_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_3386_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_3387_one__power2,axiom,
    aa_nat_real(power_power_real(one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_3388_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_3389_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_3390_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_3391_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N2)))) ) ).

% diff_le_diff_pow
tff(fact_3392_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_3393_power2__sum,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3394_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] : power_8040749407984259932d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(power_8040749407984259932d_enat(X,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),power_8040749407984259932d_enat(Y,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3395_power2__sum,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3396_power2__sum,axiom,
    ! [X: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3397_power2__sum,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3398_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_sum
tff(fact_3399_sum__squares__bound,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3400_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_squares_bound
tff(fact_3401_nat__induct2,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N: nat] :
              ( pp(aa_nat_bool(P,N))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
         => pp(aa_nat_bool(P,N2)) ) ) ) ).

% nat_induct2
tff(fact_3402_power2__diff,axiom,
    ! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3403_power2__diff,axiom,
    ! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3404_power2__diff,axiom,
    ! [X: real,Y: real] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3405_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).

% power2_diff
tff(fact_3406_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3407_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3408_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3409_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% power2_le_imp_le
tff(fact_3410_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3411_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3412_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3413_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3414_zero__le__power2,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_3415_zero__le__power2,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_3416_zero__le__power2,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% zero_le_power2
tff(fact_3417_power2__less__0,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).

% power2_less_0
tff(fact_3418_power2__less__0,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)) ).

% power2_less_0
tff(fact_3419_power2__less__0,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)) ).

% power2_less_0
tff(fact_3420_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_3421_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_3422_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_3423_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_3424_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_3425_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( ( N2 = zero_zero_nat )
        | ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_3426_less__2__cases,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( ( N2 = zero_zero_nat )
        | ( N2 = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_3427_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_3428_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_3429_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_3430_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_3431_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_3432_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_3433_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_3434_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_3435_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_3436_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_3437_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_3438_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_3439_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_3440_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_3441_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_3442_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_3443_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),N2) ).

% Suc_nat_number_of_add
tff(fact_3444_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( Xs = Ys2 )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys2,I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3445_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_bool,Ys2: list_bool] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys2) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I3))
            <=> pp(aa_nat_bool(nth_bool(Ys2),I3)) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3446_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3447_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na2351039247863542899T_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I3),nth_VEBT_VEBT(Xs3,I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_3448_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na9137082442855928016l_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_bool] :
          ( ( size_size_list_bool(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I3),aa_nat_bool(nth_bool(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_3449_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
         => ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),X_13)) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I3),aa_nat_nat(nth_nat(Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_3450_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys2,I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_3451_nth__equalityI,axiom,
    ! [Xs: list_bool,Ys2: list_bool] :
      ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
           => ( pp(aa_nat_bool(nth_bool(Xs),I2))
            <=> pp(aa_nat_bool(nth_bool(Ys2),I2)) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_3452_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
     => ( ! [I2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_3453_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% real_arch_simple
tff(fact_3454_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% real_arch_simple
tff(fact_3455_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N))) ).

% reals_Archimedean2
tff(fact_3456_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N))) ).

% reals_Archimedean2
tff(fact_3457_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).

% mult_of_nat_commute
tff(fact_3458_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_3459_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_3460_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_3461_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_3462_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2))
    <=> ? [N4: nat] : Z2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_3463_int__Suc,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int) ).

% int_Suc
tff(fact_3464_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_3465_not__zle__0__negative,axiom,
    ! [N2: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))))) ).

% not_zle_0_negative
tff(fact_3466_negative__zless__0,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))),zero_zero_int)) ).

% negative_zless_0
tff(fact_3467_negD,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
     => ? [N: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_3468_int__of__nat__induct,axiom,
    ! [P: fun_int_bool,Z2: int] :
      ( ! [N: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N)))
     => ( ! [N: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))))
       => pp(aa_int_bool(P,Z2)) ) ) ).

% int_of_nat_induct
tff(fact_3469_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N: nat] : Z2 != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_3470_zero__neq__numeral,axiom,
    ! [N2: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,N2) ).

% zero_neq_numeral
tff(fact_3471_zero__neq__numeral,axiom,
    ! [N2: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(N2) ).

% zero_neq_numeral
tff(fact_3472_zero__neq__numeral,axiom,
    ! [N2: num] : zero_zero_complex != numera6690914467698888265omplex(N2) ).

% zero_neq_numeral
tff(fact_3473_zero__neq__numeral,axiom,
    ! [N2: num] : zero_zero_real != aa_num_real(numeral_numeral_real,N2) ).

% zero_neq_numeral
tff(fact_3474_zero__neq__numeral,axiom,
    ! [N2: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N2) ).

% zero_neq_numeral
tff(fact_3475_zero__neq__numeral,axiom,
    ! [N2: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N2) ).

% zero_neq_numeral
tff(fact_3476_nat__less__real__le,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N2)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% nat_less_real_le
tff(fact_3477_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_3478_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))))
    <=> ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_3479_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(2)
tff(fact_3480_nat__le__real__less,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).

% nat_le_real_less
tff(fact_3481_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% nat_int_comparison(3)
tff(fact_3482_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% zle_int
tff(fact_3483_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_3484_int__plus,axiom,
    ! [N2: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_3485_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))),Z2) ).

% zadd_int_left
tff(fact_3486_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N2) ).

% neg_numeral_neq_numeral
tff(fact_3487_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) != aa_num_real(numeral_numeral_real,N2) ).

% neg_numeral_neq_numeral
tff(fact_3488_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(N2) ).

% neg_numeral_neq_numeral
tff(fact_3489_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,N2) ).

% neg_numeral_neq_numeral
tff(fact_3490_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(N2) ).

% neg_numeral_neq_numeral
tff(fact_3491_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) ).

% numeral_neq_neg_numeral
tff(fact_3492_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] : aa_num_real(numeral_numeral_real,M) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)) ).

% numeral_neq_neg_numeral
tff(fact_3493_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2)) ).

% numeral_neq_neg_numeral
tff(fact_3494_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)) ).

% numeral_neq_neg_numeral
tff(fact_3495_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] : numera6620942414471956472nteger(M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)) ).

% numeral_neq_neg_numeral
tff(fact_3496_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).

% int_ops(6)
tff(fact_3497_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_3498_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_3499_Ints__of__nat,axiom,
    ! [N2: nat] : pp(aa_set_complex_bool(member_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),ring_1_Ints_complex)) ).

% Ints_of_nat
tff(fact_3500_Ints__of__nat,axiom,
    ! [N2: nat] : member_real(aa_nat_real(semiri5074537144036343181t_real,N2),ring_1_Ints_real) ).

% Ints_of_nat
tff(fact_3501_Ints__of__nat,axiom,
    ! [N2: nat] : member_rat(aa_nat_rat(semiri681578069525770553at_rat,N2),ring_1_Ints_rat) ).

% Ints_of_nat
tff(fact_3502_Ints__of__nat,axiom,
    ! [N2: nat] : pp(aa_set_int_bool(member_int(aa_nat_int(semiri1314217659103216013at_int,N2)),ring_1_Ints_int)) ).

% Ints_of_nat
tff(fact_3503_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3504_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3505_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3506_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% power2_less_imp_less
tff(fact_3507_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
    <=> ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3508_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3509_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3510_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_3511_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_3512_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sum_power2_ge_zero
tff(fact_3513_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3514_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3515_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3516_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).

% not_sum_power2_lt_zero
tff(fact_3517_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).

% not_sum_power2_lt_zero
tff(fact_3518_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)) ).

% not_sum_power2_lt_zero
tff(fact_3519_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_3520_square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% square_le_1
tff(fact_3521_square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),one_one_Code_integer))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)) ) ) ).

% square_le_1
tff(fact_3522_square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).

% square_le_1
tff(fact_3523_square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)) ) ) ).

% square_le_1
tff(fact_3524_zero__le__even__power_H,axiom,
    ! [A: real,N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) ).

% zero_le_even_power'
tff(fact_3525_zero__le__even__power_H,axiom,
    ! [A: rat,N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) ).

% zero_le_even_power'
tff(fact_3526_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) ).

% zero_le_even_power'
tff(fact_3527_power__odd__eq,axiom,
    ! [A: complex,N2: 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))),N2))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3528_power__odd__eq,axiom,
    ! [A: real,N2: nat] : aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(power_power_real(aa_nat_real(power_power_real(A),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3529_power__odd__eq,axiom,
    ! [A: rat,N2: 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))),N2))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3530_power__odd__eq,axiom,
    ! [A: nat,N2: 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))),N2))) = 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),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3531_power__odd__eq,axiom,
    ! [A: int,N2: 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))),N2))) = 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),N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_3532_Ints__numeral,axiom,
    ! [N2: num] : pp(aa_set_complex_bool(member_complex(numera6690914467698888265omplex(N2)),ring_1_Ints_complex)) ).

% Ints_numeral
tff(fact_3533_Ints__numeral,axiom,
    ! [N2: num] : member_real(aa_num_real(numeral_numeral_real,N2),ring_1_Ints_real) ).

% Ints_numeral
tff(fact_3534_Ints__numeral,axiom,
    ! [N2: num] : pp(aa_set_int_bool(member_int(aa_num_int(numeral_numeral_int,N2)),ring_1_Ints_int)) ).

% Ints_numeral
tff(fact_3535_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ? [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_3536_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
       => ? [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_3537_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3538_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_3539_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3540_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3541_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3542_odd__power__less__zero,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(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))),N2)))),zero_zero_real)) ) ).

% odd_power_less_zero
tff(fact_3543_odd__power__less__zero,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),zero_zero_rat)) ) ).

% odd_power_less_zero
tff(fact_3544_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),zero_zero_int)) ) ).

% odd_power_less_zero
tff(fact_3545_power__minus1__odd,axiom,
    ! [N2: 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))),N2))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_3546_power__minus1__odd,axiom,
    ! [N2: nat] : aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_3547_power__minus1__odd,axiom,
    ! [N2: 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))),N2))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_3548_power__minus1__odd,axiom,
    ! [N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_3549_power__minus1__odd,axiom,
    ! [N2: 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))),N2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_3550_of__nat__0__le__iff,axiom,
    ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% of_nat_0_le_iff
tff(fact_3551_of__nat__0__le__iff,axiom,
    ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% of_nat_0_le_iff
tff(fact_3552_of__nat__0__le__iff,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N2))) ).

% of_nat_0_le_iff
tff(fact_3553_of__nat__0__le__iff,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N2))) ).

% of_nat_0_le_iff
tff(fact_3554_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).

% of_nat_less_0_iff
tff(fact_3555_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).

% of_nat_less_0_iff
tff(fact_3556_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).

% of_nat_less_0_iff
tff(fact_3557_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).

% of_nat_less_0_iff
tff(fact_3558_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
         => pp(aa_int_bool(P,X3)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_3559_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_3560_all__set__conv__all__nth,axiom,
    ! [Xs: list_bool,P: fun_bool_bool] :
      ( ! [X3: bool] :
          ( member_bool(X3,set_bool2(Xs))
         => pp(aa_bool_bool(P,X3)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_3561_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
         => pp(aa_nat_bool(P,X3)) )
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_3562_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_bool,X: complex] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3451745648224563538omplex(Xs)))
         => pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),I2))) )
     => ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
       => pp(aa_complex_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3563_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_bool,X: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
         => pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I2))) )
     => ( member_real(X,set_real2(Xs))
       => pp(aa_real_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3564_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_bool,X: set_nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs)))
         => pp(aa_set_nat_bool(P,aa_nat_set_nat(nth_set_nat(Xs),I2))) )
     => ( member_set_nat(X,set_set_nat2(Xs))
       => pp(aa_set_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3565_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I2))) )
     => ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3566_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
         => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I2))) )
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
       => pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3567_all__nth__imp__all__set,axiom,
    ! [Xs: list_bool,P: fun_bool_bool,X: bool] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
         => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I2))) )
     => ( member_bool(X,set_bool2(Xs))
       => pp(aa_bool_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3568_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I2))) )
     => ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3569_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3451745648224563538omplex(Xs)))
          & ( aa_nat_complex(nth_complex(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3570_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_real(Xs)))
          & ( aa_nat_real(nth_real(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3571_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs)))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3572_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3573_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs)))
          & ( nth_VEBT_VEBT(Xs,I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3574_in__set__conv__nth,axiom,
    ! [X: bool,Xs: list_bool] :
      ( member_bool(X,set_bool2(Xs))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_bool(Xs)))
          & ( pp(aa_nat_bool(nth_bool(Xs),I3))
          <=> pp(X) ) ) ) ).

% in_set_conv_nth
tff(fact_3575_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3576_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_int(Xs)))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
           => pp(aa_int_bool(P,X4)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N2))) ) ) ).

% list_ball_nth
tff(fact_3577_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s6755466524823107622T_VEBT(Xs)))
     => ( ! [X4: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
           => pp(aa_VEBT_VEBT_bool(P,X4)) )
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N2))) ) ) ).

% list_ball_nth
tff(fact_3578_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_bool,P: fun_bool_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_bool(Xs)))
     => ( ! [X4: bool] :
            ( member_bool(X4,set_bool2(Xs))
           => pp(aa_bool_bool(P,X4)) )
       => pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N2))) ) ) ).

% list_ball_nth
tff(fact_3579_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
           => pp(aa_nat_bool(P,X4)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N2))) ) ) ).

% list_ball_nth
tff(fact_3580_nth__mem,axiom,
    ! [N2: nat,Xs: list_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s3451745648224563538omplex(Xs)))
     => pp(aa_set_complex_bool(member_complex(aa_nat_complex(nth_complex(Xs),N2)),set_complex2(Xs))) ) ).

% nth_mem
tff(fact_3581_nth__mem,axiom,
    ! [N2: nat,Xs: list_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_real(Xs)))
     => member_real(aa_nat_real(nth_real(Xs),N2),set_real2(Xs)) ) ).

% nth_mem
tff(fact_3582_nth__mem,axiom,
    ! [N2: nat,Xs: list_set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s3254054031482475050et_nat(Xs)))
     => member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),N2),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_3583_nth__mem,axiom,
    ! [N2: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N2)),set_int2(Xs))) ) ).

% nth_mem
tff(fact_3584_nth__mem,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N2)),set_VEBT_VEBT2(Xs))) ) ).

% nth_mem
tff(fact_3585_nth__mem,axiom,
    ! [N2: nat,Xs: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_bool(Xs)))
     => member_bool(aa_nat_bool(nth_bool(Xs),N2),set_bool2(Xs)) ) ).

% nth_mem
tff(fact_3586_nth__mem,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N2)),set_nat2(Xs))) ) ).

% nth_mem
tff(fact_3587_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N2)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_3588_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_3589_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_3590_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N2)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_3591_of__nat__neq__0,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_3592_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_imp_less
tff(fact_3593_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_imp_less
tff(fact_3594_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_imp_less
tff(fact_3595_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% of_nat_less_imp_less
tff(fact_3596_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2))) ) ).

% less_imp_of_nat_less
tff(fact_3597_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2))) ) ).

% less_imp_of_nat_less
tff(fact_3598_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))) ) ).

% less_imp_of_nat_less
tff(fact_3599_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))) ) ).

% less_imp_of_nat_less
tff(fact_3600_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).

% of_nat_mono
tff(fact_3601_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).

% of_nat_mono
tff(fact_3602_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).

% of_nat_mono
tff(fact_3603_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).

% of_nat_mono
tff(fact_3604_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(N2)),zero_z5237406670263579293d_enat)) ).

% not_numeral_le_zero
tff(fact_3605_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N2)),zero_zero_real)) ).

% not_numeral_le_zero
tff(fact_3606_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),zero_zero_rat)) ).

% not_numeral_le_zero
tff(fact_3607_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N2)),zero_zero_nat)) ).

% not_numeral_le_zero
tff(fact_3608_not__numeral__le__zero,axiom,
    ! [N2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),zero_zero_int)) ).

% not_numeral_le_zero
tff(fact_3609_zero__le__numeral,axiom,
    ! [N2: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N2))) ).

% zero_le_numeral
tff(fact_3610_zero__le__numeral,axiom,
    ! [N2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,N2))) ).

% zero_le_numeral
tff(fact_3611_zero__le__numeral,axiom,
    ! [N2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N2))) ).

% zero_le_numeral
tff(fact_3612_zero__le__numeral,axiom,
    ! [N2: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N2))) ).

% zero_le_numeral
tff(fact_3613_zero__le__numeral,axiom,
    ! [N2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N2))) ).

% zero_le_numeral
tff(fact_3614_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),zero_zero_rat)) ).

% not_numeral_less_zero
tff(fact_3615_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N2)),zero_z5237406670263579293d_enat)) ).

% not_numeral_less_zero
tff(fact_3616_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N2)),zero_zero_real)) ).

% not_numeral_less_zero
tff(fact_3617_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N2)),zero_zero_nat)) ).

% not_numeral_less_zero
tff(fact_3618_not__numeral__less__zero,axiom,
    ! [N2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N2)),zero_zero_int)) ).

% not_numeral_less_zero
tff(fact_3619_zero__less__numeral,axiom,
    ! [N2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,N2))) ).

% zero_less_numeral
tff(fact_3620_zero__less__numeral,axiom,
    ! [N2: num] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(N2))) ).

% zero_less_numeral
tff(fact_3621_zero__less__numeral,axiom,
    ! [N2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,N2))) ).

% zero_less_numeral
tff(fact_3622_zero__less__numeral,axiom,
    ! [N2: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N2))) ).

% zero_less_numeral
tff(fact_3623_zero__less__numeral,axiom,
    ! [N2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N2))) ).

% zero_less_numeral
tff(fact_3624_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M5)),X)),C)) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_3625_one__le__numeral,axiom,
    ! [N2: num] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(N2))) ).

% one_le_numeral
tff(fact_3626_one__le__numeral,axiom,
    ! [N2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,N2))) ).

% one_le_numeral
tff(fact_3627_one__le__numeral,axiom,
    ! [N2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,N2))) ).

% one_le_numeral
tff(fact_3628_one__le__numeral,axiom,
    ! [N2: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N2))) ).

% one_le_numeral
tff(fact_3629_one__le__numeral,axiom,
    ! [N2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N2))) ).

% one_le_numeral
tff(fact_3630_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_3631_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ? [N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_3632_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% pos_int_cases
tff(fact_3633_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_3634_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% int_cases3
tff(fact_3635_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% neg_int_cases
tff(fact_3636_not__numeral__less__one,axiom,
    ! [N2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),one_one_rat)) ).

% not_numeral_less_one
tff(fact_3637_not__numeral__less__one,axiom,
    ! [N2: num] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(N2)),one_on7984719198319812577d_enat)) ).

% not_numeral_less_one
tff(fact_3638_not__numeral__less__one,axiom,
    ! [N2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N2)),one_one_real)) ).

% not_numeral_less_one
tff(fact_3639_not__numeral__less__one,axiom,
    ! [N2: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N2)),one_one_nat)) ).

% not_numeral_less_one
tff(fact_3640_not__numeral__less__one,axiom,
    ! [N2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N2)),one_one_int)) ).

% not_numeral_less_one
tff(fact_3641_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)))) ).

% not_numeral_le_neg_numeral
tff(fact_3642_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)))) ).

% not_numeral_le_neg_numeral
tff(fact_3643_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))) ).

% not_numeral_le_neg_numeral
tff(fact_3644_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)))) ).

% not_numeral_le_neg_numeral
tff(fact_3645_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N2))) ).

% neg_numeral_le_numeral
tff(fact_3646_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N2))) ).

% neg_numeral_le_numeral
tff(fact_3647_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2))) ).

% neg_numeral_le_numeral
tff(fact_3648_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2))) ).

% neg_numeral_le_numeral
tff(fact_3649_zero__neq__neg__numeral,axiom,
    ! [N2: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) ).

% zero_neq_neg_numeral
tff(fact_3650_zero__neq__neg__numeral,axiom,
    ! [N2: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)) ).

% zero_neq_neg_numeral
tff(fact_3651_zero__neq__neg__numeral,axiom,
    ! [N2: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2)) ).

% zero_neq_neg_numeral
tff(fact_3652_zero__neq__neg__numeral,axiom,
    ! [N2: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)) ).

% zero_neq_neg_numeral
tff(fact_3653_zero__neq__neg__numeral,axiom,
    ! [N2: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)) ).

% zero_neq_neg_numeral
tff(fact_3654_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)))) ).

% not_numeral_less_neg_numeral
tff(fact_3655_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)))) ).

% not_numeral_less_neg_numeral
tff(fact_3656_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))) ).

% not_numeral_less_neg_numeral
tff(fact_3657_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)))) ).

% not_numeral_less_neg_numeral
tff(fact_3658_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2))) ).

% neg_numeral_less_numeral
tff(fact_3659_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,N2))) ).

% neg_numeral_less_numeral
tff(fact_3660_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,N2))) ).

% neg_numeral_less_numeral
tff(fact_3661_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N2))) ).

% neg_numeral_less_numeral
tff(fact_3662_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_3663_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(X)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_3664_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_3665_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_3666_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_3667_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_3668_numeral__neq__neg__one,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,N2) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_3669_numeral__neq__neg__one,axiom,
    ! [N2: num] : aa_num_real(numeral_numeral_real,N2) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_3670_numeral__neq__neg__one,axiom,
    ! [N2: num] : numera6690914467698888265omplex(N2) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_3671_numeral__neq__neg__one,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,N2) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_3672_numeral__neq__neg__one,axiom,
    ! [N2: num] : numera6620942414471956472nteger(N2) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_3673_one__neq__neg__numeral,axiom,
    ! [N2: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) ).

% one_neq_neg_numeral
tff(fact_3674_one__neq__neg__numeral,axiom,
    ! [N2: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)) ).

% one_neq_neg_numeral
tff(fact_3675_one__neq__neg__numeral,axiom,
    ! [N2: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N2)) ).

% one_neq_neg_numeral
tff(fact_3676_one__neq__neg__numeral,axiom,
    ! [N2: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)) ).

% one_neq_neg_numeral
tff(fact_3677_one__neq__neg__numeral,axiom,
    ! [N2: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)) ).

% one_neq_neg_numeral
tff(fact_3678_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N2 )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_3679_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N2) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
             => ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                     => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_3680_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N2 )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_3681_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_3682_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ? [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),X))) ) ).

% ex_less_of_nat_mult
tff(fact_3683_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N2)) ) ) ).

% of_nat_diff
tff(fact_3684_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N2)) ) ) ).

% of_nat_diff
tff(fact_3685_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N2)) ) ) ).

% of_nat_diff
tff(fact_3686_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2)) ) ) ).

% of_nat_diff
tff(fact_3687_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ).

% of_nat_diff
tff(fact_3688_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)))) ).

% not_zero_le_neg_numeral
tff(fact_3689_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)))) ).

% not_zero_le_neg_numeral
tff(fact_3690_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))) ).

% not_zero_le_neg_numeral
tff(fact_3691_not__zero__le__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)))) ).

% not_zero_le_neg_numeral
tff(fact_3692_neg__numeral__le__zero,axiom,
    ! [N2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))),zero_zero_real)) ).

% neg_numeral_le_zero
tff(fact_3693_neg__numeral__le__zero,axiom,
    ! [N2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))),zero_z3403309356797280102nteger)) ).

% neg_numeral_le_zero
tff(fact_3694_neg__numeral__le__zero,axiom,
    ! [N2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))),zero_zero_rat)) ).

% neg_numeral_le_zero
tff(fact_3695_neg__numeral__le__zero,axiom,
    ! [N2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))),zero_zero_int)) ).

% neg_numeral_le_zero
tff(fact_3696_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)))) ).

% not_zero_less_neg_numeral
tff(fact_3697_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2)))) ).

% not_zero_less_neg_numeral
tff(fact_3698_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2)))) ).

% not_zero_less_neg_numeral
tff(fact_3699_not__zero__less__neg__numeral,axiom,
    ! [N2: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2)))) ).

% not_zero_less_neg_numeral
tff(fact_3700_neg__numeral__less__zero,axiom,
    ! [N2: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))),zero_zero_int)) ).

% neg_numeral_less_zero
tff(fact_3701_neg__numeral__less__zero,axiom,
    ! [N2: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))),zero_zero_real)) ).

% neg_numeral_less_zero
tff(fact_3702_neg__numeral__less__zero,axiom,
    ! [N2: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))),zero_zero_rat)) ).

% neg_numeral_less_zero
tff(fact_3703_neg__numeral__less__zero,axiom,
    ! [N2: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))),zero_z3403309356797280102nteger)) ).

% neg_numeral_less_zero
tff(fact_3704_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_le_one
tff(fact_3705_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_le_one
tff(fact_3706_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_le_one
tff(fact_3707_neg__numeral__le__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_le_one
tff(fact_3708_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_le_numeral
tff(fact_3709_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_le_numeral
tff(fact_3710_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_le_numeral
tff(fact_3711_neg__one__le__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_le_numeral
tff(fact_3712_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))) ).

% neg_numeral_le_neg_one
tff(fact_3713_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% neg_numeral_le_neg_one
tff(fact_3714_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% neg_numeral_le_neg_one
tff(fact_3715_neg__numeral__le__neg__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))) ).

% neg_numeral_le_neg_one
tff(fact_3716_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_le_neg_one
tff(fact_3717_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_le_neg_one
tff(fact_3718_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_le_neg_one
tff(fact_3719_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_le_neg_one
tff(fact_3720_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_le_neg_numeral
tff(fact_3721_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_le_neg_numeral
tff(fact_3722_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_le_neg_numeral
tff(fact_3723_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_le_neg_numeral
tff(fact_3724_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).

% neg_numeral_less_one
tff(fact_3725_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real)) ).

% neg_numeral_less_one
tff(fact_3726_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat)) ).

% neg_numeral_less_one
tff(fact_3727_neg__numeral__less__one,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).

% neg_numeral_less_one
tff(fact_3728_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).

% neg_one_less_numeral
tff(fact_3729_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M))) ).

% neg_one_less_numeral
tff(fact_3730_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M))) ).

% neg_one_less_numeral
tff(fact_3731_neg__one__less__numeral,axiom,
    ! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).

% neg_one_less_numeral
tff(fact_3732_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).

% not_numeral_less_neg_one
tff(fact_3733_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real))) ).

% not_numeral_less_neg_one
tff(fact_3734_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat))) ).

% not_numeral_less_neg_one
tff(fact_3735_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).

% not_numeral_less_neg_one
tff(fact_3736_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_one_less_neg_numeral
tff(fact_3737_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_one_less_neg_numeral
tff(fact_3738_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_one_less_neg_numeral
tff(fact_3739_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_one_less_neg_numeral
tff(fact_3740_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_3741_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_3742_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_3743_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).

% not_neg_one_less_neg_numeral
tff(fact_3744_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
         => pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N2) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M) )
             => ( ! [I2: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),X_13))
                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
                       => ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I2 )
                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2))) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I2 )
                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X4))
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma)) ) ) ) ) )
                       => pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_3745_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_3746_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_3747_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ? [R3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
          & ( aa_nat_real(power_power_real(R3),aa_nat_nat(suc,N2)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3748_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero_nat) ).

% prod_decode_def
tff(fact_3749_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
    <=> ( ( ? [A5: bool,B5: bool] : A1 = vEBT_Leaf(A5,B5)
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
               => pp(aa_nat_bool(vEBT_invar_vebt(X3),N4)) )
            & pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4)))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4))))
               => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),X_13))
                <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4))))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4))) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I3 )
                          & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4))) )
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X3))
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3750_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
     => ( ( ? [A3: bool,B3: bool] : A1 = vEBT_Leaf(A3,B3)
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_invar_vebt(X2),N)) )
                 => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                   => ( ( 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))),M5) )
                     => ( ( M5 = N )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M5) )
                         => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                                 => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_invar_vebt(X2),N)) )
                   => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                     => ( ( 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))),M5) )
                       => ( ( M5 = aa_nat_nat(suc,N) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M5) )
                           => ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                                   => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_invar_vebt(X2),N)) )
                     => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                       => ( ( 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))),M5) )
                         => ( ( M5 = N )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M5) )
                             => ( ! [I4: nat] :
                                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                   => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_13))
                                    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                                       => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                 => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N))) )
                                                & ! [X2: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I4 )
                                                      & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N))) )
                                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X2))
                                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                           => pp(aa_nat_bool(vEBT_invar_vebt(X2),N)) )
                       => ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M5))
                         => ( ( 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))),M5) )
                           => ( ( M5 = aa_nat_nat(suc,N) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M5) )
                               => ( ! [I4: nat] :
                                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                     => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),X_13))
                                      <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                                         => ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M5)))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N) = I4 )
                                                   => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N))) )
                                                  & ! [X2: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I4 )
                                                        & pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N))) )
                                                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X2))
                                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3751_in__children__def,axiom,
    ! [N2: nat,TreeList2: list_VEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N2,TreeList2),X))
    <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N2))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N2))) ) ).

% in_children_def
tff(fact_3752_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3753_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q4: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q4)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),aa_num_nat(numeral_numeral_nat,L))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2))
       => ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q4)),R2) ) ) ) ).

% divmod_step_eq
tff(fact_3754_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q4: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q4,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q4)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2))
       => ( unique5024387138958732305ep_int(L,product_Pair_int_int(Q4,R2)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q4),R2) ) ) ) ).

% divmod_step_eq
tff(fact_3755_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
            & pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% member_inv
tff(fact_3756_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
     => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),X))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3757_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N2))
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
          & ? [M5: nat] :
              ( ( some_nat(M5) = vEBT_vebt_mint(Summary) )
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),aa_nat_nat(aa_nat_fun_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,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ) ) ).

% mintlistlength
tff(fact_3758_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
       => ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info2,Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_3759_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_3760_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_3761_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_3762_high__def,axiom,
    ! [X: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% high_def
tff(fact_3763_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_3764_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_3765_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_3766_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_3767_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_3768_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_3769_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_3770_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_3771_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_3772_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_3773_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_3774_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_3775_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_3776_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_3777_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_3778_div__by__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_3779_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_3780_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_3781_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_3782_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_3783_div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_3784_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_3785_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_3786_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C) ).

% times_divide_eq_right
tff(fact_3787_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_3788_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_3789_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),B) ).

% divide_divide_eq_right
tff(fact_3790_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_3791_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_3792_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_3793_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_3794_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_3795_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C) ).

% times_divide_eq_left
tff(fact_3796_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_3797_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_3798_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_3799_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_3800_div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% div_by_1
tff(fact_3801_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_3802_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_3803__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_3804_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_3805_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_3806_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_3807_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_3808_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_3809_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_3810_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_3811_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_3812_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_3813_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_3814_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_3815_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_3816_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_3817_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_3818_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_3819_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_3820_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_3821_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_3822_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_3823_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_3824_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_3825_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_3826_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_3827_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_3828_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_3829_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_3830_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_3831_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_3832_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_3833_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_3834_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_3835_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_3836_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_3837_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_3838_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_3839_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ) ).

% divide_self_if
tff(fact_3840_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ) ).

% divide_self_if
tff(fact_3841_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ) ).

% divide_self_if
tff(fact_3842_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_3843_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_3844_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_3845_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_3846_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_3847_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_3848_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_3849_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_3850_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_3851_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_3852_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(divide_divide_int(A),A) = one_one_int ) ) ).

% div_self
tff(fact_3853_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_3854_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_3855_divide__minus1,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_3856_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_3857_divide__minus1,axiom,
    ! [X: rat] : aa_rat_rat(divide_divide_rat(X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ).

% divide_minus1
tff(fact_3858_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_3859_negative__zle,axiom,
    ! [N2: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% negative_zle
tff(fact_3860_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% zero_le_divide_1_iff
tff(fact_3861_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% zero_le_divide_1_iff
tff(fact_3862_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% divide_le_0_1_iff
tff(fact_3863_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% divide_le_0_1_iff
tff(fact_3864_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% zero_less_divide_1_iff
tff(fact_3865_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% zero_less_divide_1_iff
tff(fact_3866_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_3867_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% less_divide_eq_1_pos
tff(fact_3868_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_3869_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% less_divide_eq_1_neg
tff(fact_3870_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_3871_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% divide_less_eq_1_pos
tff(fact_3872_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_3873_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% divide_less_eq_1_neg
tff(fact_3874_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% divide_less_0_1_iff
tff(fact_3875_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% divide_less_0_1_iff
tff(fact_3876_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_3877_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% le_divide_eq_numeral1(1)
tff(fact_3878_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_3879_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_3880_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) = A )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)) ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_3881_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) = A )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_3882_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) = A )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_3883_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)) = B ) )
        & ( ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_3884_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( aa_num_real(numeral_numeral_real,W2) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)) = B ) )
        & ( ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_3885_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W2)) )
    <=> ( ( ( numera6690914467698888265omplex(W2) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W2)) = B ) )
        & ( ( numera6690914467698888265omplex(W2) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_3886_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_3887_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2)))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_3888_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W2))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W2))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_3889_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W2))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W2))),B)) ) ).

% less_divide_eq_numeral1(1)
tff(fact_3890_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_3891_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_3892_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_3893_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_3894_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_3895_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_3896_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),N2)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_3897_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_num_real(numeral_numeral_real,W2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_num_nat(numeral_numeral_nat,W2))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_3898_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N2)),aa_nat_real(semiri5074537144036343181t_real,M)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N2)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_3899_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ).

% zle_add1_eq_le
tff(fact_3900_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),one_one_int)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% zle_diff1_eq
tff(fact_3901_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_3902_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% divide_le_eq_1_neg
tff(fact_3903_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_3904_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% divide_le_eq_1_pos
tff(fact_3905_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_3906_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).

% le_divide_eq_1_neg
tff(fact_3907_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_3908_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).

% le_divide_eq_1_pos
tff(fact_3909_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_3910_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_3911_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3912_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B)) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3913_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = A )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3914_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = A )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3915_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = A )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3916_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> ( ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = B ) )
        & ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3917_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) )
    <=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = B ) )
        & ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3918_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> ( ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = B ) )
        & ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3919_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3920_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),B)) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3921_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_3922_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_3923_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_3924_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_3925_set__decode__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N2)),aa_nat_set_nat(nat_set_decode,X)))
    <=> pp(aa_set_nat_bool(member_nat(N2),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% set_decode_Suc
tff(fact_3926_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_3927_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_3928_negative__zle__0,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),zero_zero_int)) ).

% negative_zle_0
tff(fact_3929_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% not_int_zless_negative
tff(fact_3930_int__cases2,axiom,
    ! [Z2: int] :
      ( ! [N: nat] : Z2 != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% int_cases2
tff(fact_3931_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_3932_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X))) ).

% real_minus_mult_self_le
tff(fact_3933_q__pos__lemma,axiom,
    ! [B4: int,Q5: int,R4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R4)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).

% q_pos_lemma
tff(fact_3934_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N2) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3935_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B4: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R4)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B4))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3936_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B4: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R4) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q5)),R4)),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3937_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q4)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3938_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q4),Q5)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3939_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3940_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_3941_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X2: int] :
              ( pp(aa_int_bool(P,X2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3942_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_3943_odd__nonzero,axiom,
    ! [Z2: 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),Z2)),Z2) != zero_zero_int ).

% odd_nonzero
tff(fact_3944_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_3945_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_3946_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_3947_int__if,axiom,
    ! [P: bool,A: nat,B: nat] :
      ( ( pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,A) ) )
      & ( ~ pp(P)
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(P),A),B)) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ) ).

% int_if
tff(fact_3948_int__int__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N2) )
    <=> ( M = N2 ) ) ).

% int_int_eq
tff(fact_3949_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_gr_induct
tff(fact_3950_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2))
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
tff(fact_3951_less__int__code_I1_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_int_code(1)
tff(fact_3952_less__eq__int__code_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).

% less_eq_int_code(1)
tff(fact_3953_zle__iff__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2))
    <=> ? [N4: nat] : Z2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_3954_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X2: real] : member_real(X2,S3)
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( member_real(X4,S3)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z3)) )
       => ? [Y3: real] :
            ( ! [X2: real] :
                ( member_real(X2,S3)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X2),Y3)) )
            & ! [Z3: real] :
                ( ! [X4: real] :
                    ( member_real(X4,S3)
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Z3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),Z3)) ) ) ) ) ).

% complete_real
tff(fact_3955_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_3956_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X))
      | ( A = X )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A)) ) ).

% verit_la_generic
tff(fact_3957_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_3958_imp__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
           => pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
           => pp(P2) ) ) ) ) ).

% imp_le_cong
tff(fact_3959_conj__le__cong,axiom,
    ! [X: int,X6: int,P: bool,P2: bool] :
      ( ( X = X6 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
            & pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X6))
            & pp(P2) ) ) ) ) ).

% conj_le_cong
tff(fact_3960_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% int_one_le_iff_zero_less
tff(fact_3961_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z2)) ) ).

% zless_imp_add1_zle
tff(fact_3962_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)),Z2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),zero_zero_int)) ) ).

% odd_less_0_iff
tff(fact_3963_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2))) ) ).

% le_imp_0_less
tff(fact_3964_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_ge_induct
tff(fact_3965_add1__zle__eq,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% add1_zle_eq
tff(fact_3966_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
     => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_less_induct
tff(fact_3967_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M5: nat,N: nat] : Z2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M5)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_3968_plusinfinity,axiom,
    ! [D: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P2,X4))
          <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P2,X4)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P2,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_3969_minusinfinity,axiom,
    ! [D: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P1,X4))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
             => ( pp(aa_int_bool(P,X4))
              <=> pp(aa_int_bool(P1,X4)) ) )
         => ( ? [X_12: int] : pp(aa_int_bool(P1,X_12))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_3970_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int] :
            ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
         => ! [X2: int] :
              ( pp(aa_int_bool(P,X2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3971_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_3972_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
     => ( pp(aa_int_bool(P,K))
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_le_induct
tff(fact_3973_int__induct,axiom,
    ! [P: fun_int_bool,K: int,I: int] :
      ( pp(aa_int_bool(P,K))
     => ( ! [I2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
           => ( pp(aa_int_bool(P,I2))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
       => ( ! [I2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
             => ( pp(aa_int_bool(P,I2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
         => pp(aa_int_bool(P,I)) ) ) ) ).

% int_induct
tff(fact_3974_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),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,W2),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z22)) ).

% int_distrib(4)
tff(fact_3975_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: 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)),W2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W2)) ).

% int_distrib(3)
tff(fact_3976_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: 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)),W2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W2)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W2)) ).

% int_distrib(1)
tff(fact_3977_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W2),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,W2),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z22)) ).

% int_distrib(2)
tff(fact_3978_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N2) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_3979_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N2) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_3980_minus__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_3981_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_3982_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_3983_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_3984_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(Z2),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),W2)),aa_rat_rat(times_times_rat(Y),Z2)) ).

% divide_divide_times_eq
tff(fact_3985_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z2),W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),W2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2)) ).

% divide_divide_times_eq
tff(fact_3986_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z2),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W2)),aa_complex_complex(times_times_complex(Y),Z2)) ).

% divide_divide_times_eq
tff(fact_3987_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(Z2),W2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(Y),W2)) ).

% times_divide_times_eq
tff(fact_3988_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z2),W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W2)) ).

% times_divide_times_eq
tff(fact_3989_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z2),W2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(Y),W2)) ).

% times_divide_times_eq
tff(fact_3990_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_3991_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_3992_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_3993_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_3994_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_3995_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_3996_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_3997_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_3998_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_3999_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_4000_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_4001_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ).

% minus_divide_divide
tff(fact_4002_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_4003_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_4004_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_4005_int__ops_I3_J,axiom,
    ! [N2: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N2)) = aa_num_int(numeral_numeral_int,N2) ).

% int_ops(3)
tff(fact_4006_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_4007_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).

% divide_le_0_iff
tff(fact_4008_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_right_mono
tff(fact_4009_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_right_mono
tff(fact_4010_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).

% zero_le_divide_iff
tff(fact_4011_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).

% zero_le_divide_iff
tff(fact_4012_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_4013_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_nonneg
tff(fact_4014_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_nonpos
tff(fact_4015_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_nonpos
tff(fact_4016_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_nonneg
tff(fact_4017_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_nonneg
tff(fact_4018_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_4019_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_nonpos
tff(fact_4020_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_4021_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C))) ) ) ).

% divide_right_mono_neg
tff(fact_4022_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_neg_neg
tff(fact_4023_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_neg_neg
tff(fact_4024_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_neg_pos
tff(fact_4025_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_neg_pos
tff(fact_4026_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_pos_neg
tff(fact_4027_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_pos_neg
tff(fact_4028_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_pos_pos
tff(fact_4029_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_pos_pos
tff(fact_4030_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_4031_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).

% divide_less_0_iff
tff(fact_4032_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_4033_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_4034_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).

% zero_less_divide_iff
tff(fact_4035_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).

% zero_less_divide_iff
tff(fact_4036_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_4037_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono
tff(fact_4038_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_4039_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).

% divide_strict_right_mono_neg
tff(fact_4040_frac__eq__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X),Y) = aa_rat_rat(divide_divide_rat(W2),Z2) )
        <=> ( aa_rat_rat(times_times_rat(X),Z2) = aa_rat_rat(times_times_rat(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4041_frac__eq__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z2) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4042_frac__eq__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W2),Z2) )
        <=> ( aa_complex_complex(times_times_complex(X),Z2) = aa_complex_complex(times_times_complex(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4043_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
tff(fact_4044_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
tff(fact_4045_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
tff(fact_4046_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
tff(fact_4047_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
tff(fact_4048_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
tff(fact_4049_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_4050_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_4051_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_4052_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_4053_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_4054_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_4055_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_4056_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_4057_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_4058_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_4059_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_4060_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_4061_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_4062_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_4063_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_4064_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_4065_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_4066_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_4067_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_4068_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_4069_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_4070_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_4071_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_4072_power__one__over,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),N2) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),N2)) ).

% power_one_over
tff(fact_4073_power__one__over,axiom,
    ! [A: real,N2: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),N2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(A),N2)) ).

% power_one_over
tff(fact_4074_power__one__over,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),N2) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),N2)) ).

% power_one_over
tff(fact_4075_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_4076_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Z2)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_le
tff(fact_4077_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W2),Z2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_4078_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W2),Z2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z2)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less
tff(fact_4079_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W2),Z2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_4080_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W2),Z2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z2)),aa_rat_rat(divide_divide_rat(Y),W2))) ) ) ) ) ).

% frac_less2
tff(fact_4081_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_4082_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% divide_le_cancel
tff(fact_4083_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonneg_neg
tff(fact_4084_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonneg_neg
tff(fact_4085_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_4086_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonneg_pos
tff(fact_4087_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_4088_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% divide_nonpos_neg
tff(fact_4089_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).

% divide_nonpos_pos
tff(fact_4090_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),zero_zero_rat)) ) ) ).

% divide_nonpos_pos
tff(fact_4091_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_4092_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(divide_divide_int(A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_4093_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) ) ) ).

% div_positive
tff(fact_4094_div__positive,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B))) ) ) ).

% div_positive
tff(fact_4095_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_4096_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_4097_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_4098_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_less_eq
tff(fact_4099_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_4100_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq
tff(fact_4101_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_4102_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_less_eq
tff(fact_4103_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_4104_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_divide_eq
tff(fact_4105_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_4106_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_less_eq
tff(fact_4107_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_4108_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_less_divide_eq
tff(fact_4109_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(times_times_rat(Z2),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_less
tff(fact_4110_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_less
tff(fact_4111_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(Z2),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_4112_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_less_div_pos
tff(fact_4113_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_4114_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono
tff(fact_4115_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_4116_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_4117_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_4118_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% less_divide_eq_1
tff(fact_4119_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_4120_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_4121_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4122_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W2) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4123_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W2) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4124_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4125_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_num_real(numeral_numeral_real,W2) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4126_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W2) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W2)),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( numera6690914467698888265omplex(W2) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4127_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z2)),B) = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z2)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_4128_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2)),B) = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2)),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),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_4129_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B) = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_4130_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_4131_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z2)) = 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),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_4132_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_4133_add__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_4134_add__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_4135_add__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_4136_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y)),Z2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z2),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4137_add__frac__num,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4138_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z2: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z2),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4139_add__num__frac,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z2),aa_rat_rat(divide_divide_rat(X),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z2),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4140_add__num__frac,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4141_add__num__frac,axiom,
    ! [Y: complex,Z2: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z2),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4142_add__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z2)),Y)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_4143_add__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),Y)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_4144_add__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z2)),Y)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_4145_divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Z2)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Y),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_4146_divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z2)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_4147_divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z2)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_4148_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4149_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z2)) = 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),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4150_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z2)) = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z2)),B)),Z2) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_4151_diff__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_4152_diff__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_4153_diff__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(W2),Y))),aa_complex_complex(times_times_complex(Y),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_4154_diff__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),Y)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_4155_diff__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),Y)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_4156_diff__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z2)),Y)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_4157_divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z2)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(Y),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_4158_divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z2)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_4159_divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z2)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_4160_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B)) ) ).

% gt_half_sum
tff(fact_4161_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B)) ) ).

% gt_half_sum
tff(fact_4162_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)))) ) ).

% less_half_sum
tff(fact_4163_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)))) ) ).

% less_half_sum
tff(fact_4164_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4165_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4166_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_4167_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4168_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4169_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_4170_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_4171_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_4172_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4173_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_4174_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_4175_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4176_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_4177_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_4178_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4179_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) ) ) ).

% nat_mult_div_cancel1
tff(fact_4180_two__realpow__ge__one,axiom,
    ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N2))) ).

% two_realpow_ge_one
tff(fact_4181_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: real,A: real,B: real] :
      ( pp(aa_real_bool(nO_MATCH_nat_real(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4182_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4183_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4184_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( pp(aa_int_bool(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4185_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: real,A: real,B: real] :
      ( pp(aa_real_bool(nO_MATCH_int_real(aa_int_int(divide_divide_int(X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4186_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(nO_MATCH_int_rat(aa_int_int(divide_divide_int(X),Y)),C))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4187_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(nO_MATCH_int_nat(aa_int_int(divide_divide_int(X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4188_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,aa_int_int(divide_divide_int(X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4189_distrib__right__NO__MATCH,axiom,
    ! [X: real,Y: real,C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(nO_MATCH_real_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4190_distrib__right__NO__MATCH,axiom,
    ! [X: real,Y: real,C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(nO_MATCH_real_rat(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),C))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4191_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: real,B: real,C: real] :
      ( pp(aa_real_bool(nO_MATCH_nat_real(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4192_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4193_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4194_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( pp(aa_int_bool(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4195_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: real,B: real,C: real] :
      ( pp(aa_real_bool(nO_MATCH_int_real(aa_int_int(divide_divide_int(X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4196_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(nO_MATCH_int_rat(aa_int_int(divide_divide_int(X),Y)),A))
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4197_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(nO_MATCH_int_nat(aa_int_int(divide_divide_int(X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4198_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,aa_int_int(divide_divide_int(X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4199_distrib__left__NO__MATCH,axiom,
    ! [X: real,Y: real,A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(nO_MATCH_real_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4200_distrib__left__NO__MATCH,axiom,
    ! [X: real,Y: real,A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(nO_MATCH_real_rat(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A))
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4201_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: real,B: real,C: real] :
      ( pp(aa_real_bool(nO_MATCH_nat_real(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4202_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4203_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( pp(aa_int_bool(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4204_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: real,B: real,C: real] :
      ( pp(aa_real_bool(nO_MATCH_int_real(aa_int_int(divide_divide_int(X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4205_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(nO_MATCH_int_rat(aa_int_int(divide_divide_int(X),Y)),A))
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4206_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,aa_int_int(divide_divide_int(X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4207_right__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(nO_MATCH_real_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4208_right__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(nO_MATCH_real_rat(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A))
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4209_right__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: int,B: int,C: int] :
      ( pp(aa_int_bool(nO_MATCH_real_int(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4210_right__diff__distrib__NO__MATCH,axiom,
    ! [X: complex,Y: complex,A: real,B: real,C: real] :
      ( pp(aa_real_bool(nO_MAT9165723751696580935x_real(aa_complex_complex(divide1717551699836669952omplex(X),Y)),A))
     => ( 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_NO_MATCH
tff(fact_4211_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: real,A: real,B: real] :
      ( pp(aa_real_bool(nO_MATCH_nat_real(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4212_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4213_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( pp(aa_int_bool(nO_MATCH_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4214_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: real,A: real,B: real] :
      ( pp(aa_real_bool(nO_MATCH_int_real(aa_int_int(divide_divide_int(X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4215_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(nO_MATCH_int_rat(aa_int_int(divide_divide_int(X),Y)),C))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4216_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(nO_MATCH_int_int,aa_int_int(divide_divide_int(X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4217_left__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(nO_MATCH_real_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4218_left__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(nO_MATCH_real_rat(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),C))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4219_left__diff__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,C: int,A: int,B: int] :
      ( pp(aa_int_bool(nO_MATCH_real_int(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4220_left__diff__distrib__NO__MATCH,axiom,
    ! [X: complex,Y: complex,C: real,A: real,B: real] :
      ( pp(aa_real_bool(nO_MAT9165723751696580935x_real(aa_complex_complex(divide1717551699836669952omplex(X),Y)),C))
     => ( 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_NO_MATCH
tff(fact_4221_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_4222_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono_neg
tff(fact_4223_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y)),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_4224_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),Y)),X))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),aa_rat_rat(divide_divide_rat(X),Y))) ) ) ).

% mult_imp_le_div_pos
tff(fact_4225_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Y)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_le
tff(fact_4226_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(times_times_rat(Z2),Y)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),Z2)) ) ) ).

% mult_imp_div_pos_le
tff(fact_4227_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_4228_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% pos_le_divide_eq
tff(fact_4229_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_4230_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_divide_le_eq
tff(fact_4231_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_4232_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_divide_eq
tff(fact_4233_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_4234_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).

% neg_divide_le_eq
tff(fact_4235_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).

% divide_left_mono
tff(fact_4236_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B))) ) ) ) ).

% divide_left_mono
tff(fact_4237_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_4238_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq
tff(fact_4239_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_4240_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% divide_le_eq
tff(fact_4241_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_4242_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_4243_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_4244_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% le_divide_eq_1
tff(fact_4245_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4246_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4247_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4248_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4249_frac__le__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),zero_zero_real)) ) ) ) ).

% frac_le_eq
tff(fact_4250_frac__le__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z2)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z2))),zero_zero_rat)) ) ) ) ).

% frac_le_eq
tff(fact_4251_frac__less__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y)),aa_rat_rat(divide_divide_rat(W2),Z2)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W2),Y))),aa_rat_rat(times_times_rat(Y),Z2))),zero_zero_rat)) ) ) ) ).

% frac_less_eq
tff(fact_4252_frac__less__eq,axiom,
    ! [Y: real,Z2: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z2)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),zero_zero_real)) ) ) ) ).

% frac_less_eq
tff(fact_4253_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4254_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4255_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4256_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4257_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4258_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4259_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4260_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4261_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_4262_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_4263_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_4264_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_4265_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> ( ( ( C != zero_zero_real )
         => ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4266_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( ( C != zero_zero_complex )
         => ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4267_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> ( ( ( C != zero_zero_rat )
         => ( B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4268_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> ( ( ( C != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) = B ) )
        & ( ( C = zero_zero_real )
         => ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4269_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( ( C != zero_zero_complex )
         => ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))),C) = B ) )
        & ( ( C = zero_zero_complex )
         => ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4270_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( ( C != zero_zero_rat )
         => ( aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) = B ) )
        & ( ( C = zero_zero_rat )
         => ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4271_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z2))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4272_minus__divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z2))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4273_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z2))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4274_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2))),B) = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2))),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),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4275_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B) = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4276_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z2))),B) = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z2))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4277_field__sum__of__halves,axiom,
    ! [X: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_4278_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_4279_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2))),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2))),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),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4280_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4281_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z2))),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z2))),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4282_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2)),B) = aa_real_real(uminus_uminus_real,B) ) )
      & ( ( Z2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z2)),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),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4283_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4284_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z2)),B) = aa_rat_rat(uminus_uminus_rat,B) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z2)),B) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z2))),Z2) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4285_minus__divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z2))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4286_minus__divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z2))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4287_minus__divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z2))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4288_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N2)) ) ) ) ).

% power_diff
tff(fact_4289_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ) ).

% power_diff
tff(fact_4290_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N2)) ) ) ) ).

% power_diff
tff(fact_4291_power__diff,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_real(power_power_real(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N2)) ) ) ) ).

% power_diff
tff(fact_4292_power__diff,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N2)) ) ) ) ).

% power_diff
tff(fact_4293_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ) ) ) ).

% div_geq
tff(fact_4294_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W2)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4295_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W2)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4296_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4297_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4298_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_4299_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_4300_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_4301_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_4302_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4303_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4304_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4305_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4306_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4307_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B))) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4308_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4309_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4310_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% half_gt_zero_iff
tff(fact_4311_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% half_gt_zero_iff
tff(fact_4312_half__gt__zero,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_4313_half__gt__zero,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% half_gt_zero
tff(fact_4314_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),S))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_4315_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R2))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R2),S))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V)) ) ) ) ).

% scaling_mono
tff(fact_4316_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4317_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4318_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4319_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4320_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_4321_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% field_less_half_sum
tff(fact_4322_nat__approx__posE,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ~ ! [N: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E2)) ) ).

% nat_approx_posE
tff(fact_4323_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
     => ~ ! [N: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E2)) ) ).

% nat_approx_posE
tff(fact_4324_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( ( N2 != zero_zero_nat )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N2)))) ) ) ).

% inverse_of_nat_le
tff(fact_4325_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( ( N2 != zero_zero_nat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,N2)))) ) ) ).

% inverse_of_nat_le
tff(fact_4326_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4327_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4328_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),B)) )
        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
         => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C))) )
            & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4329_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),aa_rat_rat(divide_divide_rat(B),C)))
    <=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),B)) )
        & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
         => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C))) )
            & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4330_triangle__def,axiom,
    ! [N2: nat] : aa_nat_nat(nat_triangle,N2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_4331_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( aa_nat_real(power_power_real(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_4332_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(X),Y) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ) ) ) ).

% arith_geo_mean
tff(fact_4333_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_4334_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_4335_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_4336_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_4337_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_4338_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_4339_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,Va2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),N2))
     => ( ( N2 = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).

% nested_mint
tff(fact_4340_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),N2) = M ) ) ).

% div_mult_self_is_m
tff(fact_4341_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_4342_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_4343_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_4344_bits__div__0,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_4345_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_4346_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(divide_divide_int(A),one_one_int) = A ).

% bits_div_by_1
tff(fact_4347_zdiv__numeral__Bit0,axiom,
    ! [V: num,W2: 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,W2))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W2)) ).

% zdiv_numeral_Bit0
tff(fact_4348_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)),Deg))
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_4349_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_4350_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% div_mult_mult1_if
tff(fact_4351_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_4352_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_4353_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_4354_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_4355_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_4356_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_4357_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_4358_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat ) ) ).

% div_less
tff(fact_4359_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_4360_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_4361_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_4362_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_4363_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_4364_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_4365_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_4366_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_4367_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_4368_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_4369_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M)),N2) = M ) ) ).

% div_mult_self1_is_m
tff(fact_4370_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% half_nonnegative_int_iff
tff(fact_4371_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% half_negative_int_iff
tff(fact_4372_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_4373_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_neg_pos_less0
tff(fact_4374_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_4375_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_4376_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_4377_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A4),B))) ) ) ).

% zdiv_mono1
tff(fact_4378_zdiv__mono2,axiom,
    ! [A: int,B4: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),aa_int_int(divide_divide_int(A),B4))) ) ) ) ).

% zdiv_mono2
tff(fact_4379_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(divide_divide_int(I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_4380_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A4),B)),aa_int_int(divide_divide_int(A),B))) ) ) ).

% zdiv_mono1_neg
tff(fact_4381_zdiv__mono2__neg,axiom,
    ! [A: int,B4: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B4),B))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B4)),aa_int_int(divide_divide_int(A),B))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_4382_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(K),L)))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).

% div_int_pos_iff
tff(fact_4383_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(K),L))) ) ) ).

% div_positive_int
tff(fact_4384_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_4385_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(A),B)),zero_zero_int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_4386_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(I),K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_4387_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_4388_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_4389_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(divide_divide_int(A),B)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_4390_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(X),K)),X)) ) ) ).

% int_div_less_self
tff(fact_4391_real__of__nat__div4,axiom,
    ! [N2: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(semiri5074537144036343181t_real,X)))) ).

% real_of_nat_div4
tff(fact_4392_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N2)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(divide_divide_int(B2),N2)),aa_int_int(divide_divide_int(A2),N2))) ) ) ).

% verit_less_mono_div_int2
tff(fact_4393_div__eq__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_4394_real__of__nat__div2,axiom,
    ! [N2: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),X))))) ).

% real_of_nat_div2
tff(fact_4395_real__of__nat__div3,axiom,
    ! [N2: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),X)))),one_one_real)) ).

% real_of_nat_div3
tff(fact_4396_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_4397_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: 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),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_4398_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: 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),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(divide_divide_int(A),B) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_4399_split__zdiv,axiom,
    ! [P: fun_int_bool,N2: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(divide_divide_int(N2),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,I3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,I3)) ) ) ) ) ).

% split_zdiv
tff(fact_4400_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
       => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4401_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_4402_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),K))) ) ).

% div_le_mono
tff(fact_4403_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),M)) ).

% div_le_dividend
tff(fact_4404_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_4405_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(divide_divide_int(B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_4406_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_4407_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N2))) ).

% Suc_div_le_mono
tff(fact_4408_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),I)) ) ).

% less_mult_imp_div_less
tff(fact_4409_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),N2)),M)) ).

% div_times_less_eq_dividend
tff(fact_4410_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))),M)) ).

% times_div_less_eq_dividend
tff(fact_4411_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_4412_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_4413_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_4414_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_4415_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),zero_zero_int)) ).

% not_exp_less_eq_0_int
tff(fact_4416_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% div_greater_zero_iff
tff(fact_4417_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).

% div_le_mono2
tff(fact_4418_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q4)),N2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4))) ) ) ).

% div_less_iff_less_mult
tff(fact_4419_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),M)) ) ) ).

% div_less_dividend
tff(fact_4420_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = M )
      <=> ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_4421_div__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
          | ( N2 = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
            | ( N2 = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ) ) ) ).

% div_if
tff(fact_4422_div__nat__eqI,axiom,
    ! [N2: nat,Q4: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,Q4))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_4423_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q4))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),Q4)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),N2)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_4424_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2))))) ) ).

% dividend_less_times_div
tff(fact_4425_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),N2)))) ) ).

% dividend_less_div_times
tff(fact_4426_split__div,axiom,
    ! [P: fun_nat_bool,M: nat,N2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)))
    <=> ( ( ( N2 = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N2))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I3)),J3) )
               => pp(aa_nat_bool(P,I3)) ) ) ) ) ) ).

% split_div
tff(fact_4427_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) != 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))),N2) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4428_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4429_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_4430_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_4431_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) != 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,N2),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4432_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) != 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,N2),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4433_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ).

% div_exp_eq
tff(fact_4434_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: nat] : aa_int_int(divide_divide_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ).

% div_exp_eq
tff(fact_4435_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) = 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,N2),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4436_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N2)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
         => ( aa_int_int(divide_divide_int(aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N2)) = 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,N2),M))) ) ) ) ) ).

% power_diff_power_eq
tff(fact_4437_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2)) ) ) ) ).

% le_div_geq
tff(fact_4438_split__div_H,axiom,
    ! [P: fun_nat_bool,M: nat,N2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)))
    <=> ( ( ( N2 = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q6)),M))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,Q6))))
            & pp(aa_nat_bool(P,Q6)) ) ) ) ).

% split_div'
tff(fact_4439_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(P,N))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) )
       => ( ! [N: nat] :
              ( pp(aa_nat_bool(P,N))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) )
         => pp(aa_nat_bool(P,N2)) ) ) ) ).

% nat_bit_induct
tff(fact_4440_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_4441_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_4442_int__bit__induct,axiom,
    ! [P: fun_int_bool,K: int] :
      ( pp(aa_int_bool(P,zero_zero_int))
     => ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
       => ( ! [K2: int] :
              ( pp(aa_int_bool(P,K2))
             => ( ( K2 != zero_zero_int )
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
         => ( ! [K2: int] :
                ( pp(aa_int_bool(P,K2))
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => pp(aa_int_bool(P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_4443_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_4444_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
               => ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                   => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_4445_inrange,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => pp(aa_set_nat_bool(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),one_one_nat)))) ) ).

% inrange
tff(fact_4446_linear__plus__1__le__power,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X)),one_one_real)),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),N2))) ) ).

% linear_plus_1_le_power
tff(fact_4447_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_4448_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_4449_enat__ord__number_I1_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ) ).

% enat_ord_number(1)
tff(fact_4450_enat__ord__number_I2_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ) ).

% enat_ord_number(2)
tff(fact_4451_i0__less,axiom,
    ! [N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N2))
    <=> ( N2 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_4452_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
          & ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4453_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
          & ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4454_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
          & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4455_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
          & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4456_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
          & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4457_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
          & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),H2)) ) ) ) ).

% Icc_eq_Icc
tff(fact_4458_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(L),I))
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4459_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( member_rat(I,set_or633870826150836451st_rat(L,U))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4460_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( member_num(I,set_or7049704709247886629st_num(L,U))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4461_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4462_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_int(L,U)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4463_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( member_real(I,set_or1222579329274155063t_real(L,U))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),U)) ) ) ).

% atLeastAtMost_iff
tff(fact_4464_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_4465_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N2),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% set_bit_negative_int_iff
tff(fact_4466_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))) ).

% finite_atLeastAtMost
tff(fact_4467_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( set_or5403411693681687835d_enat(A,B) = bot_bo7653980558646680370d_enat )
    <=> ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4468_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4469_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4470_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4471_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4472_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4473_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff
tff(fact_4474_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = set_or5403411693681687835d_enat(A,B) )
    <=> ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4475_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4476_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4477_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4478_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4479_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4480_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% atLeastatMost_empty_iff2
tff(fact_4481_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))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
        | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
          & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4482_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))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4483_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))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
        | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
          & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4484_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4485_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4486_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))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4487_atLeastatMost__empty,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
     => ( set_or5403411693681687835d_enat(A,B) = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
tff(fact_4488_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_4489_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4490_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4491_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4492_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4493_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ finite_finite_rat(set_or633870826150836451st_rat(A,B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% infinite_Icc_iff
tff(fact_4494_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ finite_finite_real(set_or1222579329274155063t_real(A,B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% infinite_Icc_iff
tff(fact_4495_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I))
     => ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4496_list__update__beyond,axiom,
    ! [Xs: list_bool,I: nat,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I))
     => ( list_update_bool(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4497_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I))
     => ( list_update_nat(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4498_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4499_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),I))
      <=> pp(X) ) ) ).

% nth_list_update_eq
tff(fact_4500_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_4501_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_4502_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_4503_set__swap,axiom,
    ! [I: nat,Xs: list_bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))) = set_bool2(Xs) ) ) ) ).

% set_swap
tff(fact_4504_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_4505_not__iless0,axiom,
    ! [N2: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N2),zero_z5237406670263579293d_enat)) ).

% not_iless0
tff(fact_4506_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N2),zero_z5237406670263579293d_enat))
    <=> ( N2 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_4507_i0__lb,axiom,
    ! [N2: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N2)) ).

% i0_lb
tff(fact_4508_enat__less__induct,axiom,
    ! [P: fun_Ex8563670519792538868t_bool,N2: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M2),N))
             => pp(aa_Ext2772979874139974381t_bool(P,M2)) )
         => pp(aa_Ext2772979874139974381t_bool(P,N)) )
     => pp(aa_Ext2772979874139974381t_bool(P,N2)) ) ).

% enat_less_induct
tff(fact_4509_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N2)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N2)) ) ) ).

% enat_0_less_mult_iff
tff(fact_4510_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ~ finite_finite_rat(set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_4511_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ~ finite_finite_real(set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_4512_ex__nat__less,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ? [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N2))
          & pp(aa_nat_bool(P,M6)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less
tff(fact_4513_all__nat__less,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N2))
         => pp(aa_nat_bool(P,M6)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less
tff(fact_4514_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,X: complex,I: nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),A2))
     => ( pp(aa_set_complex_bool(member_complex(X),A2))
       => pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4515_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( ord_less_eq_set_real(set_real2(Xs),A2)
     => ( member_real(X,A2)
       => ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_4516_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),A2)
     => ( member_set_nat(X,A2)
       => ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_4517_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A2))
     => ( pp(aa_set_int_bool(member_int(X),A2))
       => pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4518_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X)),A2) ) ) ).

% set_update_subsetI
tff(fact_4519_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),A2))
     => ( pp(aa_set_nat_bool(member_nat(X),A2))
       => pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X))),A2)) ) ) ).

% set_update_subsetI
tff(fact_4520_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))
    <=> ( ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
          | ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
            & pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
            & ( pp(aa_set_nat_bool(ord_less_set_nat(C),A))
              | pp(aa_set_nat_bool(ord_less_set_nat(B),D)) ) ) )
        & pp(aa_set_nat_bool(ord_less_eq_set_nat(C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4521_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))
    <=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
          | ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D))
            & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
              | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),D)) ) ) )
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4522_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))
    <=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
          | ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
            & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D))
            & ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
              | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),D)) ) ) )
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4523_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
    <=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D))
            & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
              | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),D)) ) ) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4524_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D)))
    <=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
          | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D))
            & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
              | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),D)) ) ) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4525_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))
    <=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
          | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
            & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
              | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4526_set__update__memI,axiom,
    ! [N2: nat,Xs: list_complex,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s3451745648224563538omplex(Xs)))
     => pp(aa_set_complex_bool(member_complex(X),set_complex2(list_update_complex(Xs,N2,X)))) ) ).

% set_update_memI
tff(fact_4527_set__update__memI,axiom,
    ! [N2: nat,Xs: list_real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_real(Xs)))
     => member_real(X,set_real2(list_update_real(Xs,N2,X))) ) ).

% set_update_memI
tff(fact_4528_set__update__memI,axiom,
    ! [N2: nat,Xs: list_set_nat,X: set_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s3254054031482475050et_nat(Xs)))
     => member_set_nat(X,set_set_nat2(list_update_set_nat(Xs,N2,X))) ) ).

% set_update_memI
tff(fact_4529_set__update__memI,axiom,
    ! [N2: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_int(Xs)))
     => pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(Xs,N2,X)))) ) ).

% set_update_memI
tff(fact_4530_set__update__memI,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s6755466524823107622T_VEBT(Xs)))
     => pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N2,X)))) ) ).

% set_update_memI
tff(fact_4531_set__update__memI,axiom,
    ! [N2: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_bool(Xs)))
     => member_bool(X,set_bool2(list_update_bool(Xs,N2,X))) ) ).

% set_update_memI
tff(fact_4532_set__update__memI,axiom,
    ! [N2: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
     => pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(Xs,N2,X)))) ) ).

% set_update_memI
tff(fact_4533_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4534_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( ( list_update_bool(Xs,I,X) = Xs )
      <=> ( pp(aa_nat_bool(nth_bool(Xs),I))
        <=> pp(X) ) ) ) ).

% list_update_same_conv
tff(fact_4535_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_4536_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
     => ( ( ( I = J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = X ) )
        & ( ( I != J )
         => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_4537_nth__list__update,axiom,
    ! [I: nat,Xs: list_bool,X: bool,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
     => ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),J))
      <=> ( ( ( I = J )
           => pp(X) )
          & ( ( I != J )
           => pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).

% nth_list_update
tff(fact_4538_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_4539_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4540_Bolzano,axiom,
    ! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [A3: real,B3: real,C3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B3),C3))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),B3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),C3))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),C3)) ) ) ) )
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
               => ? [D3: real] :
                    ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D3))
                    & ! [A3: real,B3: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A3),X4))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B3))
                          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A3)),D3)) )
                       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A3),B3)) ) ) ) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).

% Bolzano
tff(fact_4541_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_4542_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_4543_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_4544_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_4545_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_less(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_4546_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_less(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% succ_greatereq_min
tff(fact_4547_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_greater(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% pred_lesseq_max
tff(fact_4548_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_greater(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_4549_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).

% set_vebt'_def
tff(fact_4550_idiff__0,axiom,
    ! [N2: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat),N2) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_4551_idiff__0__right,axiom,
    ! [N2: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N2),zero_z5237406670263579293d_enat) = N2 ).

% idiff_0_right
tff(fact_4552_finite__interval__int1,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_af(A),B)))) ).

% finite_interval_int1
tff(fact_4553_finite__interval__int4,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ag(A),B)))) ).

% finite_interval_int4
tff(fact_4554_succ__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_succ(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah(T),X)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_4555_pred__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(T),N2))
     => ( ( vEBT_vebt_pred(T,X) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ai(T),X)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_4556_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_4557_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N2),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% unset_bit_negative_int_iff
tff(fact_4558_max__bot,axiom,
    ! [X: set_Extended_enat] : ord_ma4205026669011143323d_enat(bot_bo7653980558646680370d_enat,X) = X ).

% max_bot
tff(fact_4559_max__bot,axiom,
    ! [X: set_real] : ord_max_set_real(bot_bot_set_real,X) = X ).

% max_bot
tff(fact_4560_max__bot,axiom,
    ! [X: set_nat] : ord_max_set_nat(bot_bot_set_nat,X) = X ).

% max_bot
tff(fact_4561_max__bot,axiom,
    ! [X: set_int] : ord_max_set_int(bot_bot_set_int,X) = X ).

% max_bot
tff(fact_4562_max__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),X) = X ).

% max_bot
tff(fact_4563_max__bot,axiom,
    ! [X: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(bot_bo4199563552545308370d_enat),X) = X ).

% max_bot
tff(fact_4564_max__bot2,axiom,
    ! [X: set_Extended_enat] : ord_ma4205026669011143323d_enat(X,bot_bo7653980558646680370d_enat) = X ).

% max_bot2
tff(fact_4565_max__bot2,axiom,
    ! [X: set_real] : ord_max_set_real(X,bot_bot_set_real) = X ).

% max_bot2
tff(fact_4566_max__bot2,axiom,
    ! [X: set_nat] : ord_max_set_nat(X,bot_bot_set_nat) = X ).

% max_bot2
tff(fact_4567_max__bot2,axiom,
    ! [X: set_int] : ord_max_set_int(X,bot_bot_set_int) = X ).

% max_bot2
tff(fact_4568_max__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),bot_bot_nat) = X ).

% max_bot2
tff(fact_4569_max__bot2,axiom,
    ! [X: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),bot_bo4199563552545308370d_enat) = X ).

% max_bot2
tff(fact_4570_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)) ).

% max_Suc_Suc
tff(fact_4571_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_4572_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_4573_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_4574_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_4575_max__0L,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N2) = N2 ).

% max_0L
tff(fact_4576_max__0R,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),zero_zero_nat) = N2 ).

% max_0R
tff(fact_4577_finite__interval__int3,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_aj(A),B)))) ).

% finite_interval_int3
tff(fact_4578_finite__interval__int2,axiom,
    ! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_ak(A),B)))) ).

% finite_interval_int2
tff(fact_4579_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,K)))) ).

% finite_Collect_less_nat
tff(fact_4580_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_am,K)))) ).

% finite_Collect_le_nat
tff(fact_4581_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(1)
tff(fact_4582_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).

% max_number_of(1)
tff(fact_4583_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(1)
tff(fact_4584_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(1)
tff(fact_4585_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,V) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).

% max_number_of(1)
tff(fact_4586_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(1)
tff(fact_4587_max__0__1_I4_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(4)
tff(fact_4588_max__0__1_I4_J,axiom,
    ! [X: num] : ord_max_Code_integer(numera6620942414471956472nteger(X),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(X) ).

% max_0_1(4)
tff(fact_4589_max__0__1_I4_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(4)
tff(fact_4590_max__0__1_I4_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),zero_zero_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(4)
tff(fact_4591_max__0__1_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(4)
tff(fact_4592_max__0__1_I4_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X)),zero_zero_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(4)
tff(fact_4593_max__0__1_I3_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(3)
tff(fact_4594_max__0__1_I3_J,axiom,
    ! [X: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,numera6620942414471956472nteger(X)) = numera6620942414471956472nteger(X) ).

% max_0_1(3)
tff(fact_4595_max__0__1_I3_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(3)
tff(fact_4596_max__0__1_I3_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(3)
tff(fact_4597_max__0__1_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(3)
tff(fact_4598_max__0__1_I3_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(3)
tff(fact_4599_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_4600_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_4601_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_4602_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_4603_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_4604_max__0__1_I1_J,axiom,
    ord_max_Code_integer(zero_z3403309356797280102nteger,one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_4605_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_4606_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_4607_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_4608_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_4609_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_4610_max__0__1_I2_J,axiom,
    ord_max_Code_integer(one_one_Code_integer,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_4611_max__0__1_I6_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(6)
tff(fact_4612_max__0__1_I6_J,axiom,
    ! [X: num] : ord_max_Code_integer(numera6620942414471956472nteger(X),one_one_Code_integer) = numera6620942414471956472nteger(X) ).

% max_0_1(6)
tff(fact_4613_max__0__1_I6_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).

% max_0_1(6)
tff(fact_4614_max__0__1_I6_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,X)),one_one_real) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(6)
tff(fact_4615_max__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(6)
tff(fact_4616_max__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X)),one_one_int) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(6)
tff(fact_4617_max__0__1_I5_J,axiom,
    ! [X: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_num_rat(numeral_numeral_rat,X) ).

% max_0_1(5)
tff(fact_4618_max__0__1_I5_J,axiom,
    ! [X: num] : ord_max_Code_integer(one_one_Code_integer,numera6620942414471956472nteger(X)) = numera6620942414471956472nteger(X) ).

% max_0_1(5)
tff(fact_4619_max__0__1_I5_J,axiom,
    ! [X: num] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).

% max_0_1(5)
tff(fact_4620_max__0__1_I5_J,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_num_real(numeral_numeral_real,X) ).

% max_0_1(5)
tff(fact_4621_max__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).

% max_0_1(5)
tff(fact_4622_max__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).

% max_0_1(5)
tff(fact_4623_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_num_real(numeral_numeral_real,U) ) ) ) ).

% max_number_of(2)
tff(fact_4624_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = numera6620942414471956472nteger(U) ) ) ) ).

% max_number_of(2)
tff(fact_4625_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_num_rat(numeral_numeral_rat,U) ) ) ) ).

% max_number_of(2)
tff(fact_4626_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_num_int(numeral_numeral_int,U) ) ) ) ).

% max_number_of(2)
tff(fact_4627_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_num_real(numeral_numeral_real,V) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4628_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(3)
tff(fact_4629_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_num_rat(numeral_numeral_rat,V) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4630_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(3)
tff(fact_4631_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))))
       => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4632_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
       => ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).

% max_number_of(4)
tff(fact_4633_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))))
       => ( aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4634_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).

% max_number_of(4)
tff(fact_4635_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                   => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                      & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_4636_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
               => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(Sn),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(Sn)))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_4637_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)),none_nat)),Mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_4638_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = 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_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_nu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at_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)) ) ) ) ) ).

% del_in_range
tff(fact_4639_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),H))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L)),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_4640_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                        & ( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H))))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_4641_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList2,H,Newnode) )
                     => ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,Newnode))
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xn),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(Sn),none_nat)),Xn),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(Sn)))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(vEBT_vebt_maxt(Sn)))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_4642_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary),X) = if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList2,Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_4643_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_4644_max__def__raw,axiom,
    ! [X2: extended_enat,Xa: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X2),Xa))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X2),Xa) = Xa ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X2),Xa))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_4645_max__def__raw,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X2),Xa))
       => ( ord_max_Code_integer(X2,Xa) = Xa ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X2),Xa))
       => ( ord_max_Code_integer(X2,Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_4646_max__def__raw,axiom,
    ! [X2: set_nat,Xa: set_nat] :
      ( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X2),Xa))
       => ( ord_max_set_nat(X2,Xa) = Xa ) )
      & ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(X2),Xa))
       => ( ord_max_set_nat(X2,Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_4647_max__def__raw,axiom,
    ! [X2: rat,Xa: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Xa))
       => ( aa_rat_rat(ord_max_rat(X2),Xa) = Xa ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X2),Xa))
       => ( aa_rat_rat(ord_max_rat(X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_4648_max__def__raw,axiom,
    ! [X2: num,Xa: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Xa))
       => ( aa_num_num(ord_max_num(X2),Xa) = Xa ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X2),Xa))
       => ( aa_num_num(ord_max_num(X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_4649_max__def__raw,axiom,
    ! [X2: nat,Xa: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Xa) = Xa ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_4650_max__def__raw,axiom,
    ! [X2: int,Xa: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Xa) = Xa ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Xa))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X2),Xa) = X2 ) ) ) ).

% max_def_raw
tff(fact_4651_iadd__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N2) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_4652_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_4653_less__set__def,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
    <=> ord_le3776645642988763374x_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2)) ) ).

% less_set_def
tff(fact_4654_less__set__def,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ord_less_set_real(A2,B2)
    <=> ord_le7519368973566061808l_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).

% less_set_def
tff(fact_4655_less__set__def,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ord_less_set_set_nat(A2,B2)
    <=> ord_le4523237222650602006t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2)) ) ).

% less_set_def
tff(fact_4656_less__set__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
    <=> ord_le6841235830588351564t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).

% less_set_def
tff(fact_4657_less__set__def,axiom,
    ! [A2: set_int,B2: set_int] :
      ( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
    <=> ord_le3716944864119727344t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).

% less_set_def
tff(fact_4658_imult__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N2) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_4659_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,X)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y)) ).

% of_nat_max
tff(fact_4660_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,X),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ).

% of_nat_max
tff(fact_4661_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_4662_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_rat_rat(ord_max_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(semiri681578069525770553at_rat,Y)) ).

% of_nat_max
tff(fact_4663_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_4664_of__nat__max,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_4665_lambda__zero,axiom,
    aTP_Lamm_an = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_4666_lambda__zero,axiom,
    aTP_Lamm_ao = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_4667_lambda__zero,axiom,
    aTP_Lamm_ap = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_4668_lambda__zero,axiom,
    aTP_Lamm_aq = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_4669_lambda__one,axiom,
    aTP_Lamm_ar = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_4670_lambda__one,axiom,
    aTP_Lamm_as = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_4671_lambda__one,axiom,
    aTP_Lamm_at = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_4672_lambda__one,axiom,
    aTP_Lamm_au = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_4673_lambda__one,axiom,
    aTP_Lamm_av = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_4674_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_aw(P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_4675_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N)))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ax(F),U)))) ) ).

% finite_less_ub
tff(fact_4676_max__absorb2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X),Y))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4677_max__absorb2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
     => ( ord_max_Code_integer(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_4678_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
     => ( ord_max_set_nat(X,Y) = Y ) ) ).

% max_absorb2
tff(fact_4679_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
     => ( aa_rat_rat(ord_max_rat(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4680_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( aa_num_num(ord_max_num(X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4681_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4682_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = Y ) ) ).

% max_absorb2
tff(fact_4683_max__absorb1,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Y),X))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y) = X ) ) ).

% max_absorb1
tff(fact_4684_max__absorb1,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X))
     => ( ord_max_Code_integer(X,Y) = X ) ) ).

% max_absorb1
tff(fact_4685_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X))
     => ( ord_max_set_nat(X,Y) = X ) ) ).

% max_absorb1
tff(fact_4686_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
     => ( aa_rat_rat(ord_max_rat(X),Y) = X ) ) ).

% max_absorb1
tff(fact_4687_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ( aa_num_num(ord_max_num(X),Y) = X ) ) ).

% max_absorb1
tff(fact_4688_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y) = X ) ) ).

% max_absorb1
tff(fact_4689_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y) = X ) ) ).

% max_absorb1
tff(fact_4690_max__def,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) )
      & ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
       => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ) ).

% max_def
tff(fact_4691_max__def,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
       => ( ord_max_Code_integer(A,B) = B ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
       => ( ord_max_Code_integer(A,B) = A ) ) ) ).

% max_def
tff(fact_4692_max__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ord_max_set_nat(A,B) = B ) )
      & ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
       => ( ord_max_set_nat(A,B) = A ) ) ) ).

% max_def
tff(fact_4693_max__def,axiom,
    ! [A: rat,B: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = B ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ) ).

% max_def
tff(fact_4694_max__def,axiom,
    ! [A: num,B: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(ord_max_num(A),B) = B ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
       => ( aa_num_num(ord_max_num(A),B) = A ) ) ) ).

% max_def
tff(fact_4695_max__def,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ) ).

% max_def
tff(fact_4696_max__def,axiom,
    ! [A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ) ).

% max_def
tff(fact_4697_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z2)) ).

% max_add_distrib_right
tff(fact_4698_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z2: rat] : aa_rat_rat(plus_plus_rat(X),aa_rat_rat(ord_max_rat(Y),Z2)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X),Y)),aa_rat_rat(plus_plus_rat(X),Z2)) ).

% max_add_distrib_right
tff(fact_4699_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z2)) ).

% max_add_distrib_right
tff(fact_4700_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z2)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z2)) ).

% max_add_distrib_right
tff(fact_4701_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] : plus_p5714425477246183910nteger(X,ord_max_Code_integer(Y,Z2)) = ord_max_Code_integer(plus_p5714425477246183910nteger(X,Y),plus_p5714425477246183910nteger(X,Z2)) ).

% max_add_distrib_right
tff(fact_4702_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z2) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z2)) ).

% max_add_distrib_left
tff(fact_4703_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z2) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(X),Z2)),aa_rat_rat(plus_plus_rat(Y),Z2)) ).

% max_add_distrib_left
tff(fact_4704_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z2)) ).

% max_add_distrib_left
tff(fact_4705_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z2) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z2)) ).

% max_add_distrib_left
tff(fact_4706_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] : plus_p5714425477246183910nteger(ord_max_Code_integer(X,Y),Z2) = ord_max_Code_integer(plus_p5714425477246183910nteger(X,Z2),plus_p5714425477246183910nteger(Y,Z2)) ).

% max_add_distrib_left
tff(fact_4707_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] : minus_8373710615458151222nteger(ord_max_Code_integer(X,Y),Z2) = ord_max_Code_integer(minus_8373710615458151222nteger(X,Z2),minus_8373710615458151222nteger(Y,Z2)) ).

% max_diff_distrib_left
tff(fact_4708_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z2) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z2)) ).

% max_diff_distrib_left
tff(fact_4709_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(X),Y)),Z2) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(X),Z2)),aa_rat_rat(minus_minus_rat(Y),Z2)) ).

% max_diff_distrib_left
tff(fact_4710_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z2) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z2)) ).

% max_diff_distrib_left
tff(fact_4711_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ).

% nat_add_max_right
tff(fact_4712_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),Q4)) ).

% nat_add_max_left
tff(fact_4713_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_max_right
tff(fact_4714_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ).

% nat_mult_max_left
tff(fact_4715_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).

% set_vebt_def
tff(fact_4716_numeral__code_I2_J,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N2)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N2)),aa_num_rat(numeral_numeral_rat,N2)) ).

% numeral_code(2)
tff(fact_4717_numeral__code_I2_J,axiom,
    ! [N2: num] : numera1916890842035813515d_enat(aa_num_num(bit0,N2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N2)),numera1916890842035813515d_enat(N2)) ).

% numeral_code(2)
tff(fact_4718_numeral__code_I2_J,axiom,
    ! [N2: num] : numera6690914467698888265omplex(aa_num_num(bit0,N2)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N2)),numera6690914467698888265omplex(N2)) ).

% numeral_code(2)
tff(fact_4719_numeral__code_I2_J,axiom,
    ! [N2: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N2)),aa_num_real(numeral_numeral_real,N2)) ).

% numeral_code(2)
tff(fact_4720_numeral__code_I2_J,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,N2)) ).

% numeral_code(2)
tff(fact_4721_numeral__code_I2_J,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,N2)) ).

% numeral_code(2)
tff(fact_4722_finite__int__segment,axiom,
    ! [A: real,B: real] : finite_finite_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ay(A),B))) ).

% finite_int_segment
tff(fact_4723_finite__int__segment,axiom,
    ! [A: rat,B: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_az(A),B))) ).

% finite_int_segment
tff(fact_4724_nat__less__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_less_as_int
tff(fact_4725_nat__leq__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Xa))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).

% nat_leq_as_int
tff(fact_4726_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),M) ).

% nat_minus_add_max
tff(fact_4727_finite__lists__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_ba(A2),N2))) ) ).

% finite_lists_length_eq
tff(fact_4728_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_bb(A2),N2))) ) ).

% finite_lists_length_eq
tff(fact_4729_finite__lists__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_bc(A2),N2))) ) ).

% finite_lists_length_eq
tff(fact_4730_finite__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_bd(A2),N2))) ) ).

% finite_lists_length_eq
tff(fact_4731_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_be(A2),N2))) ) ).

% finite_lists_length_eq
tff(fact_4732_finite__lists__length__eq,axiom,
    ! [A2: set_bool,N2: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bf(A2),N2))) ) ).

% finite_lists_length_eq
tff(fact_4733_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_bg(A2),N2))) ) ).

% finite_lists_length_eq
tff(fact_4734_finite__lists__length__le,axiom,
    ! [A2: set_int,N2: nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_bh(A2),N2))) ) ).

% finite_lists_length_le
tff(fact_4735_finite__lists__length__le,axiom,
    ! [A2: set_complex,N2: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_bi(A2),N2))) ) ).

% finite_lists_length_le
tff(fact_4736_finite__lists__length__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => finite500796754983035824at_nat(collec3343600615725829874at_nat(aa_nat1772366067986465313t_bool(aTP_Lamm_bj(A2),N2))) ) ).

% finite_lists_length_le
tff(fact_4737_finite__lists__length__le,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => finite1862508098717546133d_enat(collec8433460942617342167d_enat(aa_nat4017767907309459068t_bool(aTP_Lamm_bk(A2),N2))) ) ).

% finite_lists_length_le
tff(fact_4738_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_bl(A2),N2))) ) ).

% finite_lists_length_le
tff(fact_4739_finite__lists__length__le,axiom,
    ! [A2: set_bool,N2: nat] :
      ( finite_finite_bool(A2)
     => finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bm(A2),N2))) ) ).

% finite_lists_length_le
tff(fact_4740_finite__lists__length__le,axiom,
    ! [A2: set_nat,N2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_bn(A2),N2))) ) ).

% finite_lists_length_le
tff(fact_4741_aset_I2_J,axiom,
    ! [D4: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4))) ) ) ) ) ) ).

% aset(2)
tff(fact_4742_aset_I1_J,axiom,
    ! [D4: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),A2))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),A2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4))) ) ) ) ) ) ).

% aset(1)
tff(fact_4743_bset_I2_J,axiom,
    ! [D4: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                | pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4))) ) ) ) ) ) ).

% bset(2)
tff(fact_4744_bset_I1_J,axiom,
    ! [D4: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb: int] :
                  ( pp(aa_set_int_bool(member_int(Xb),B2))
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
         => ( pp(aa_int_bool(P,X4))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb: int] :
                    ( pp(aa_set_int_bool(member_int(Xb),B2))
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
           => ( pp(aa_int_bool(Q,X4))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( ( pp(aa_int_bool(P,X2))
                & pp(aa_int_bool(Q,X2)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4))) ) ) ) ) ) ).

% bset(1)
tff(fact_4745_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ! [X4: int,K2: int] :
            ( pp(aa_int_bool(P,X4))
          <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) )
       => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
        <=> ? [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D)))
              & pp(aa_int_bool(P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4746_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4))) ) ) ) ).

% aset(7)
tff(fact_4747_aset_I5_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)),T)) ) ) ) ) ).

% aset(5)
tff(fact_4748_aset_I4_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),A2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4) != T ) ) ) ) ) ).

% aset(4)
tff(fact_4749_aset_I3_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4) = T ) ) ) ) ) ).

% aset(3)
tff(fact_4750_bset_I7_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4))) ) ) ) ) ).

% bset(7)
tff(fact_4751_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)),T)) ) ) ) ).

% bset(5)
tff(fact_4752_bset_I4_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(T),B2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4) != T ) ) ) ) ) ).

% bset(4)
tff(fact_4753_bset_I3_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4) = T ) ) ) ) ) ).

% bset(3)
tff(fact_4754_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)) ).

% vebt_insert.simps(5)
tff(fact_4755_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2,S)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_4756_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4))) ) ) ) ).

% aset(8)
tff(fact_4757_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),A2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)),T)) ) ) ) ) ).

% aset(6)
tff(fact_4758_bset_I8_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
               => ! [Xb2: int] :
                    ( pp(aa_set_int_bool(member_int(Xb2),B2))
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4))) ) ) ) ) ).

% bset(8)
tff(fact_4759_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)),T)) ) ) ) ).

% bset(6)
tff(fact_4760_cpmi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,B2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(P2,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),B2))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa_int_bool(P2,X4))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),B2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4761_cppi,axiom,
    ! [D4: int,P: fun_int_bool,P2: fun_int_bool,A2: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D4))
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
           => ( pp(aa_int_bool(P,X4))
            <=> pp(aa_int_bool(P2,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4)))
                 => ! [Xb: int] :
                      ( pp(aa_set_int_bool(member_int(Xb),A2))
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Xa) ) ) )
             => ( pp(aa_int_bool(P,X4))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa_int_bool(P2,X4))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4)))) )
           => ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & pp(aa_int_bool(P2,X3)) )
                | ? [X3: int] :
                    ( pp(aa_set_int_bool(member_int(X3),set_or1266510415728281911st_int(one_one_int,D4)))
                    & ? [Xa4: int] :
                        ( pp(aa_set_int_bool(member_int(Xa4),A2))
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa4),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4762_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2,Vd)),X))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_4763_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y = vEBT_Leaf(fTrue,B3) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y = vEBT_Leaf(A3,fTrue) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,zero_zero_nat,Ts,S2) )
             => ( Y != vEBT_Node(Info,zero_zero_nat,Ts,S2) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
               => ( Y != vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                   => ( Y != if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_4764_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
                 => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
                     => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_4765_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList2,Vc)),X))
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_4766_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_4767_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
             => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                   => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_4768_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2)
               => ( pp(Y)
                <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_4769_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_4770_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => pp(A3) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => pp(B3) )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                               => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_4771_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_4772_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_4773_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( ( Xa2 = zero_zero_nat )
               => pp(A3) )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => pp(B3) )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_4774_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_4775_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = some_nat(Mi) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_less(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_4776_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = some_nat(Ma) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_greater(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_4777_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary),X) = if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_delete(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),Ma))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary))))))),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary)) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_4778_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y != vEBT_Leaf(fFalse,B3) ) ) )
       => ( ! [A3: bool] :
              ( ? [B3: bool] : X = vEBT_Leaf(A3,B3)
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf(A3,fFalse) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A3,B3) )
               => ( ? [N: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Y != vEBT_Leaf(A3,B3) ) ) )
           => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                   => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                     => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                             => ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) ) )
                            & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y = 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)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_4779_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( ! [Uu2: bool,B3: bool] :
            ( ( X = vEBT_Leaf(Uu2,B3) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( pp(B3)
                   => ( Y = some_nat(one_one_nat) ) )
                  & ( ~ pp(B3)
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
           => ( ? [N: nat] : Xa2 = aa_nat_nat(suc,N)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( Y = some_nat(Mi2) ) )
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_less(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_4780_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: bool] :
              ( ? [Uw2: bool] : X = vEBT_Leaf(A3,Uw2)
             => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
               => ~ ( ( pp(A3)
                     => ( Y = some_nat(zero_zero_nat) ) )
                    & ( ~ pp(A3)
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A3,B3) )
               => ( ? [Va: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va))
                 => ~ ( ( pp(B3)
                       => ( Y = some_nat(one_one_nat) ) )
                      & ( ~ pp(B3)
                       => ( ( pp(A3)
                           => ( Y = some_nat(zero_zero_nat) ) )
                          & ( ~ pp(A3)
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                       => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                             => ( Y = some_nat(Ma2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_greater(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_4781_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_bo(N2),C)))) ) ).

% finite_nth_roots
tff(fact_4782_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => finite_finite_real(collect_real(aTP_Lamm_bp(N2))) ) ).

% finite_roots_unity
tff(fact_4783_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bq(N2)))) ) ).

% finite_roots_unity
tff(fact_4784_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,B3: bool] :
              ( ( X = vEBT_Leaf(Uu2,B3) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( pp(B3)
                     => ( Y = some_nat(one_one_nat) ) )
                    & ( ~ pp(B3)
                     => ( Y = none_nat ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B3),zero_zero_nat)) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw2) )
               => ! [N: nat] :
                    ( ( Xa2 = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X = 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),Xa2)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xa2)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                       => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                             => ( Y = some_nat(Mi2) ) )
                            & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                             => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_less(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_4785_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(A3,Uw2) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( ( pp(A3)
                       => ( Y = some_nat(zero_zero_nat) ) )
                      & ( ~ pp(A3)
                       => ( Y = none_nat ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,Uw2),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A3,B3) )
                 => ! [Va: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( ( pp(B3)
                           => ( Y = some_nat(one_one_nat) ) )
                          & ( ~ pp(B3)
                           => ( ( pp(A3)
                               => ( Y = some_nat(zero_zero_nat) ) )
                              & ( ~ pp(A3)
                               => ( Y = none_nat ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xa2)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xa2)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                         => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( Y = some_nat(Ma2) ) )
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,fequal_option_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat))),vEBT_VEBT_greater(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(fequal_option_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_4786_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb3
tff(fact_4787_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb3
tff(fact_4788_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_4789_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_4790_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb3
tff(fact_4791_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_4792_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_4793_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(A),B))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb4
tff(fact_4794_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb4
tff(fact_4795_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_4796_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_4797_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb4
tff(fact_4798_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_4799_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_4800_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y)),Z2))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X),Z2))
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_4801_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,ord_max_Code_integer(X,Y)),Z2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z2))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_4802_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_4803_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(X),Y)),Z2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_4804_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(X),Y)),Z2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z2))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_4805_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_4806_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2)) ) ) ).

% max_less_iff_conj
tff(fact_4807_max__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(Q4),zero_z5237406670263579293d_enat) = Q4 ).

% max_enat_simps(2)
tff(fact_4808_max__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat),Q4) = Q4 ).

% max_enat_simps(3)
tff(fact_4809_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
        & pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).

% max.bounded_iff
tff(fact_4810_max_Obounded__iff,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A)) ) ) ).

% max.bounded_iff
tff(fact_4811_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_4812_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),A))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.bounded_iff
tff(fact_4813_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.bounded_iff
tff(fact_4814_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.bounded_iff
tff(fact_4815_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb2
tff(fact_4816_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb2
tff(fact_4817_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_4818_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb2
tff(fact_4819_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_4820_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_4821_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb1
tff(fact_4822_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb1
tff(fact_4823_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_4824_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb1
tff(fact_4825_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_4826_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_4827_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or1266510415728281911st_int(L,U))) ).

% finite_atLeastAtMost_int
tff(fact_4828_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(D),B))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(C),D)),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B))) ) ) ).

% max.mono
tff(fact_4829_max_Omono,axiom,
    ! [C: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,D),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(C,D)),ord_max_Code_integer(A,B))) ) ) ).

% max.mono
tff(fact_4830_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C),D)),aa_rat_rat(ord_max_rat(A),B))) ) ) ).

% max.mono
tff(fact_4831_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,D),B))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(C),D)),aa_num_num(ord_max_num(A),B))) ) ) ).

% max.mono
tff(fact_4832_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).

% max.mono
tff(fact_4833_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ) ).

% max.mono
tff(fact_4834_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ) ) ).

% max.orderE
tff(fact_4835_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( A = ord_max_Code_integer(A,B) ) ) ).

% max.orderE
tff(fact_4836_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_4837_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.orderE
tff(fact_4838_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_4839_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_4840_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) )
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A)) ) ).

% max.orderI
tff(fact_4841_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = ord_max_Code_integer(A,B) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A)) ) ).

% max.orderI
tff(fact_4842_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).

% max.orderI
tff(fact_4843_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(ord_max_num(A),B) )
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).

% max.orderI
tff(fact_4844_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).

% max.orderI
tff(fact_4845_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).

% max.orderI
tff(fact_4846_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).

% max.boundedE
tff(fact_4847_max_OboundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A))
     => ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
         => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A)) ) ) ).

% max.boundedE
tff(fact_4848_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).

% max.boundedE
tff(fact_4849_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).

% max.boundedE
tff(fact_4850_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).

% max.boundedE
tff(fact_4851_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).

% max.boundedE
tff(fact_4852_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
     => ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
       => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_4853_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A)) ) ) ).

% max.boundedI
tff(fact_4854_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_4855_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),A)) ) ) ).

% max.boundedI
tff(fact_4856_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_4857_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)) ) ) ).

% max.boundedI
tff(fact_4858_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
    <=> ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) ) ) ).

% max.order_iff
tff(fact_4859_max_Oorder__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
    <=> ( A = ord_max_Code_integer(A,B) ) ) ).

% max.order_iff
tff(fact_4860_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_4861_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.order_iff
tff(fact_4862_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_4863_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_4864_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B))) ).

% max.cobounded1
tff(fact_4865_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),ord_max_Code_integer(A,B))) ).

% max.cobounded1
tff(fact_4866_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded1
tff(fact_4867_max_Ocobounded1,axiom,
    ! [A: num,B: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(ord_max_num(A),B))) ).

% max.cobounded1
tff(fact_4868_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded1
tff(fact_4869_max_Ocobounded1,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded1
tff(fact_4870_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B))) ).

% max.cobounded2
tff(fact_4871_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),ord_max_Code_integer(A,B))) ).

% max.cobounded2
tff(fact_4872_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B))) ).

% max.cobounded2
tff(fact_4873_max_Ocobounded2,axiom,
    ! [B: num,A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),aa_num_num(ord_max_num(A),B))) ).

% max.cobounded2
tff(fact_4874_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).

% max.cobounded2
tff(fact_4875_max_Ocobounded2,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).

% max.cobounded2
tff(fact_4876_le__max__iff__disj,axiom,
    ! [Z2: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z2),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z2),X))
        | pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_4877_le__max__iff__disj,axiom,
    ! [Z2: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Z2),ord_max_Code_integer(X,Y)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Z2),X))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_4878_le__max__iff__disj,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),aa_rat_rat(ord_max_rat(X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_4879_le__max__iff__disj,axiom,
    ! [Z2: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z2),aa_num_num(ord_max_num(X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z2),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_4880_le__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_4881_le__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),Y)) ) ) ).

% le_max_iff_disj
tff(fact_4882_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
    <=> ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4883_max_Oabsorb__iff1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),A))
    <=> ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_4884_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4885_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
    <=> ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4886_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4887_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_4888_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
    <=> ( aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4889_max_Oabsorb__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
    <=> ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_4890_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4891_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
    <=> ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4892_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4893_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_4894_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B))) ) ).

% max.coboundedI1
tff(fact_4895_max_OcoboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.coboundedI1
tff(fact_4896_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI1
tff(fact_4897_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.coboundedI1
tff(fact_4898_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI1
tff(fact_4899_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI1
tff(fact_4900_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),B))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B))) ) ).

% max.coboundedI2
tff(fact_4901_max_OcoboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.coboundedI2
tff(fact_4902_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.coboundedI2
tff(fact_4903_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.coboundedI2
tff(fact_4904_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.coboundedI2
tff(fact_4905_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.coboundedI2
tff(fact_4906_less__max__iff__disj,axiom,
    ! [Z2: extended_enat,X: extended_enat,Y: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z2),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(X),Y)))
    <=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z2),X))
        | pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_4907_less__max__iff__disj,axiom,
    ! [Z2: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),ord_max_Code_integer(X,Y)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_4908_less__max__iff__disj,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_4909_less__max__iff__disj,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),aa_rat_rat(ord_max_rat(X),Y)))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_4910_less__max__iff__disj,axiom,
    ! [Z2: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),aa_num_num(ord_max_num(X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_4911_less__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_4912_less__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),Y)) ) ) ).

% less_max_iff_disj
tff(fact_4913_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(B),C)),A))
     => ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
         => ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A)) ) ) ).

% max.strict_boundedE
tff(fact_4914_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,ord_max_Code_integer(B,C)),A))
     => ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
         => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_4915_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A))
     => ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
         => ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_4916_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),A))
     => ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
         => ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_4917_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_max_num(B),C)),A))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_4918_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_4919_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).

% max.strict_boundedE
tff(fact_4920_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
    <=> ( ( A = aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4921_max_Ostrict__order__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),A))
    <=> ( ( A = ord_max_Code_integer(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4922_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4923_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4924_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
    <=> ( ( A = aa_num_num(ord_max_num(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4925_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4926_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_4927_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_4928_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.strict_coboundedI1
tff(fact_4929_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_4930_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_4931_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.strict_coboundedI1
tff(fact_4932_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_4933_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI1
tff(fact_4934_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),B))
     => pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(ord_ma741700101516333627d_enat(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_4935_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B))) ) ).

% max.strict_coboundedI2
tff(fact_4936_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_4937_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_4938_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(ord_max_num(A),B))) ) ).

% max.strict_coboundedI2
tff(fact_4939_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_4940_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).

% max.strict_coboundedI2
tff(fact_4941_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = vEBT_Leaf(fFalse,B3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),zero_zero_nat)) ) ) )
         => ( ! [A3: bool,B3: bool] :
                ( ( X = vEBT_Leaf(A3,B3) )
               => ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf(A3,fFalse) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: bool,B3: bool] :
                  ( ( X = vEBT_Leaf(A3,B3) )
                 => ! [N: nat] :
                      ( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = vEBT_Leaf(A3,B3) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList,Summary2),Xa2)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2,Smry2),Xa2)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                         => ( ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                  | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                               => ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) ) )
                              & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                    | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y = if_VEBT_VEBT(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Xa2)),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Ma2))),aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2))),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fequal_option_nat(vEBT_vebt_maxt(vEBT_vebt_delete(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),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(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_4942_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y = vEBT_Leaf(fTrue,B3) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y = vEBT_Leaf(A3,fTrue) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y = vEBT_Leaf(A3,B3) ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,zero_zero_nat,Ts,S2) )
               => ( ( Y = vEBT_Node(Info,zero_zero_nat,Ts,S2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,zero_zero_nat,Ts,S2),Xa2)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
                 => ( ( Y = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2),Xa2)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary2),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                     => ( ( Y = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4943_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ pp(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),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                                 => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                         => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4944_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                               => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                  & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                       => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4945_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4946_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),Xa2))
                 => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4947_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ( ( ( Xa2 = zero_zero_nat )
                   => pp(A3) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => pp(B3) )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S2),Xa2))
                   => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                       => pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                      & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4948_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A3,B3),Xa2))
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => pp(A3) )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => pp(B3) )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary2),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
                           => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
                               => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
                                   => pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
                                  & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4949_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = 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),Xa2))
                       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                           => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4950_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa2)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                             => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4951_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3,Vb2),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList,Vc2),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X = 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),Xa2))
                   => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),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)))))) )
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),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))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4952_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4953_lemma__termdiff3,axiom,
    ! [H: real,Z2: real,K5: real,N2: nat] :
      ( ( H != zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z2)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),H)),N2)),aa_nat_real(power_power_real(Z2),N2))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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,N2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4954_lemma__termdiff3,axiom,
    ! [H: complex,Z2: complex,K5: real,N2: nat] :
      ( ( H != zero_zero_complex )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),K5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z2),H))),K5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z2),H)),N2)),aa_nat_complex(power_power_complex(Z2),N2))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),aa_nat_complex(power_power_complex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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,N2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(power_power_real(K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).

% lemma_termdiff3
tff(fact_4955_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_4956_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_4957_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_4958_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
    <=> ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_4959_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
    <=> ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_4960_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_4961_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_4962_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_4963_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,X) = zero_zero_real )
    <=> ( X = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_4964_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_4965_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_4966_square__norm__one,axiom,
    ! [X: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_4967_square__norm__one,axiom,
    ! [X: complex] :
      ( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_4968_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_4969_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(ln_ln_real,Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_4970_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_4971_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_4972_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% ln_less_zero_iff
tff(fact_4973_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_4974_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_4975_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4976_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% ln_le_zero_iff
tff(fact_4977_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_4978_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_less_self
tff(fact_4979_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X)) ) ).

% ln_bound
tff(fact_4980_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_4981_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)) ) ) ).

% ln_less_zero
tff(fact_4982_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_gt_zero
tff(fact_4983_ln__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).

% ln_ge_zero
tff(fact_4984_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_4985_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_4986_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_4987_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_4988_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_4989_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real))) ) ).

% ln_le_minus_one
tff(fact_4990_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),Y))) ) ) ).

% ln_diff_le
tff(fact_4991_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_4992_ln__realpow,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_nat_real(power_power_real(X),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_4993_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))),aa_real_real(uminus_uminus_real,X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_4994_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_4995_norm__not__less__zero,axiom,
    ! [X: complex] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)) ).

% norm_not_less_zero
tff(fact_4996_norm__ge__zero,axiom,
    ! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X))) ).

% norm_ge_zero
tff(fact_4997_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_4998_norm__uminus__minus,axiom,
    ! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% norm_uminus_minus
tff(fact_4999_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).

% norm_uminus_minus
tff(fact_5000_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_5001_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_5002_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N2: nat,Z2: real] :
      ( ( aa_nat_real(power_power_real(W2),N2) = aa_nat_real(power_power_real(Z2),N2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5003_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N2: nat,Z2: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),N2) = aa_nat_complex(power_power_complex(Z2),N2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5004_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_5005_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S))) ) ) ).

% norm_mult_less
tff(fact_5006_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_5007_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_lt
tff(fact_5008_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_5009_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_add_less
tff(fact_5010_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C))) ) ).

% norm_add_leD
tff(fact_5011_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C))) ) ).

% norm_add_leD
tff(fact_5012_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_5013_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E2)) ) ).

% norm_triangle_le
tff(fact_5014_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).

% norm_triangle_ineq
tff(fact_5015_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).

% norm_triangle_ineq
tff(fact_5016_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_5017_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S))) ) ) ).

% norm_triangle_mono
tff(fact_5018_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z2: real,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z2))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Z2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5019_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z2: complex,E22: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z2))),E22))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).

% norm_diff_triangle_less
tff(fact_5020_norm__diff__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))) ).

% norm_diff_ineq
tff(fact_5021_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B)))) ).

% norm_diff_ineq
tff(fact_5022_power__eq__1__iff,axiom,
    ! [W2: real,N2: nat] :
      ( ( aa_nat_real(power_power_real(W2),N2) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5023_power__eq__1__iff,axiom,
    ! [W2: complex,N2: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),N2) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5024_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),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_5025_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),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_5026_ln__2__less__1,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real)) ).

% ln_2_less_1
tff(fact_5027_ln__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aTP_Lamm_br(X)) ) ) ) ).

% ln_series
tff(fact_5028_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5029_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_5030_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5031_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_5032_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5033_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_5034_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5035_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
     => ( eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B,product_Pair_int_int(Q4,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B),product_Pair_int_int(Q4,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5036_low__def,axiom,
    ! [X: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% low_def
tff(fact_5037_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_5038_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_5039_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_5040_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_5041_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_5042_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_5043_abs__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_abs
tff(fact_5044_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_5045_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_5046_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_5047_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_5048_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_5049_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_5050_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_5051_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_5052_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_5053_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_5054_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_5055_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_5056_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_5057_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_5058_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_5059_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_5060_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_5061_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_5062_mod__self,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),A) = zero_zero_int ).

% mod_self
tff(fact_5063_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_5064_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_5065_mod__by__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),zero_zero_int) = A ).

% mod_by_0
tff(fact_5066_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_5067_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_5068_mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_5069_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_5070_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_5071_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_5072_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_5073_abs__numeral,axiom,
    ! [N2: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,N2)) = aa_num_rat(numeral_numeral_rat,N2) ).

% abs_numeral
tff(fact_5074_abs__numeral,axiom,
    ! [N2: num] : abs_abs_Code_integer(numera6620942414471956472nteger(N2)) = numera6620942414471956472nteger(N2) ).

% abs_numeral
tff(fact_5075_abs__numeral,axiom,
    ! [N2: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,N2)) = aa_num_real(numeral_numeral_real,N2) ).

% abs_numeral
tff(fact_5076_abs__numeral,axiom,
    ! [N2: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N2)) = aa_num_int(numeral_numeral_int,N2) ).

% abs_numeral
tff(fact_5077_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_5078_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_5079_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_5080_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_5081_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_5082_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_5083_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_5084_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_5085_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_5086_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_5087_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_5088_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_5089_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_5090_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_5091_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_5092_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_5093_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_5094_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_5095_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_5096_abs__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_5097_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_5098_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_5099_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_5100_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_5101_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_5102_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_5103_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_5104_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_5105_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_5106_abs__minus,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus
tff(fact_5107_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_5108_abs__of__nat,axiom,
    ! [N2: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N2)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N2) ).

% abs_of_nat
tff(fact_5109_abs__of__nat,axiom,
    ! [N2: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N2)) = aa_nat_real(semiri5074537144036343181t_real,N2) ).

% abs_of_nat
tff(fact_5110_abs__of__nat,axiom,
    ! [N2: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)) = aa_nat_rat(semiri681578069525770553at_rat,N2) ).

% abs_of_nat
tff(fact_5111_abs__of__nat,axiom,
    ! [N2: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_int(semiri1314217659103216013at_int,N2) ).

% abs_of_nat
tff(fact_5112_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = M ) ) ).

% mod_less
tff(fact_5113_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_5114_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_5115_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_5116_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger))
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_5117_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_5118_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat))
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_5119_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_5120_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).

% abs_le_self_iff
tff(fact_5121_abs__le__self__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% abs_le_self_iff
tff(fact_5122_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% abs_le_self_iff
tff(fact_5123_abs__le__self__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).

% abs_le_self_iff
tff(fact_5124_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_5125_abs__of__nonneg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5126_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5127_abs__of__nonneg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_5128_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_5129_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_5130_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_5131_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_5132_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_5133_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_5134_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_5135_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_5136_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_5137_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_5138_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_5139_mod__by__1,axiom,
    ! [A: int] : aa_int_int(modulo_modulo_int(A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_5140_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_5141_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_5142_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_5143_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_5144_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_5145_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_5146_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_5147_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_5148_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_5149_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_5150_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_5151_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_5152_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_5153_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_5154_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_5155_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_5156_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_5157_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_5158_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_5159_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_5160_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_5161_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_5162_abs__neg__numeral,axiom,
    ! [N2: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_num_int(numeral_numeral_int,N2) ).

% abs_neg_numeral
tff(fact_5163_abs__neg__numeral,axiom,
    ! [N2: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_num_real(numeral_numeral_real,N2) ).

% abs_neg_numeral
tff(fact_5164_abs__neg__numeral,axiom,
    ! [N2: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_num_rat(numeral_numeral_rat,N2) ).

% abs_neg_numeral
tff(fact_5165_abs__neg__numeral,axiom,
    ! [N2: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N2))) = numera6620942414471956472nteger(N2) ).

% abs_neg_numeral
tff(fact_5166_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_5167_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_5168_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_5169_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_5170_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_5171_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_5172_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_5173_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_5174_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_5175_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_simps(1)
tff(fact_5176_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_5177_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_simps(1)
tff(fact_5178_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_5179_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5180_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5181_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5182_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5183_abs__of__nonpos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_5184_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_5185_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_5186_abs__of__nonpos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_5187_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_5188_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5189_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self1
tff(fact_5190_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self2
tff(fact_5191_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)),M))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self3
tff(fact_5192_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),K)),M))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% Suc_mod_mult_self4
tff(fact_5193_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_5194_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_5195_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_5196_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_5197_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_5198_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_5199_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_5200_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N2)))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5201_zero__less__power__abs__iff,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N2)))
    <=> ( ( A != zero_zero_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5202_zero__less__power__abs__iff,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N2)))
    <=> ( ( A != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5203_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N2)))
    <=> ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5204_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_5205_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N2: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),N2))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_5206_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_5207_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_5208_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_5209_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_5210_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_bs(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_5211_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_bt(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_5212_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_5213_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_5214_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5215_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_5216_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_5217_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5218_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_5219_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_5220_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_5221_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_5222_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_5223_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_5224_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_5225_unique__remainder,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q4,R2))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q5,R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_5226_unique__quotient,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q4,R2))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q5,R4))
       => ( Q4 = Q5 ) ) ) ).

% unique_quotient
tff(fact_5227_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).

% abs_le_D1
tff(fact_5228_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).

% abs_le_D1
tff(fact_5229_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).

% abs_le_D1
tff(fact_5230_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).

% abs_le_D1
tff(fact_5231_abs__ge__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).

% abs_ge_self
tff(fact_5232_abs__ge__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A))) ).

% abs_ge_self
tff(fact_5233_abs__ge__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_self
tff(fact_5234_abs__ge__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).

% abs_ge_self
tff(fact_5235_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5236_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_5237_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_5238_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_5239_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_5240_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_5241_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_5242_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_5243_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_5244_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_5245_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_5246_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_5247_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_5248_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_5249_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_5250_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_5251_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( X = Y )
        | ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5252_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( X = Y )
        | ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5253_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,X) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( X = Y )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5254_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(X) = abs_abs_Code_integer(Y) )
    <=> ( ( X = Y )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5255_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_5256_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_5257_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_5258_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: 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,A4),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,B4),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,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_5259_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( aa_int_int(modulo_modulo_int(A),C) = aa_int_int(modulo_modulo_int(A4),C) )
     => ( ( aa_int_int(modulo_modulo_int(B),C) = aa_int_int(modulo_modulo_int(B4),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,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_5260_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A4,B4),C) ) ) ) ).

% mod_add_cong
tff(fact_5261_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_5262_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_5263_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_5264_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_5265_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_5266_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_5267_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N2) ).

% mod_Suc_Suc_eq
tff(fact_5268_mod__Suc__eq,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) ).

% mod_Suc_eq
tff(fact_5269_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),M)) ).

% mod_less_eq_dividend
tff(fact_5270_Ints__abs,axiom,
    ! [A: int] :
      ( pp(aa_set_int_bool(member_int(A),ring_1_Ints_int))
     => pp(aa_set_int_bool(member_int(aa_int_int(abs_abs_int,A)),ring_1_Ints_int)) ) ).

% Ints_abs
tff(fact_5271_Ints__abs,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => member_rat(aa_rat_rat(abs_abs_rat,A),ring_1_Ints_rat) ) ).

% Ints_abs
tff(fact_5272_Ints__abs,axiom,
    ! [A: code_integer] :
      ( member_Code_integer(A,ring_11222124179247155820nteger)
     => member_Code_integer(abs_abs_Code_integer(A),ring_11222124179247155820nteger) ) ).

% Ints_abs
tff(fact_5273_Ints__abs,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_real_real(abs_abs_real,A),ring_1_Ints_real) ) ).

% Ints_abs
tff(fact_5274_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero_int,product_Pair_int_int(zero_zero_int,K)) ).

% eucl_rel_int_by0
tff(fact_5275_div__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q4,R2))
     => ( aa_int_int(divide_divide_int(K),L) = Q4 ) ) ).

% div_int_unique
tff(fact_5276_abs__ge__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))) ).

% abs_ge_zero
tff(fact_5277_abs__ge__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).

% abs_ge_zero
tff(fact_5278_abs__ge__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_zero
tff(fact_5279_abs__ge__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).

% abs_ge_zero
tff(fact_5280_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)) ).

% abs_not_less_zero
tff(fact_5281_abs__not__less__zero,axiom,
    ! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).

% abs_not_less_zero
tff(fact_5282_abs__not__less__zero,axiom,
    ! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)) ).

% abs_not_less_zero
tff(fact_5283_abs__not__less__zero,axiom,
    ! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).

% abs_not_less_zero
tff(fact_5284_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_5285_abs__of__pos,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_5286_abs__of__pos,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_5287_abs__of__pos,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_5288_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq
tff(fact_5289_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq
tff(fact_5290_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq
tff(fact_5291_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq
tff(fact_5292_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D))) ) ) ).

% abs_mult_less
tff(fact_5293_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B)),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).

% abs_mult_less
tff(fact_5294_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D))) ) ) ).

% abs_mult_less
tff(fact_5295_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B)),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))) ) ) ).

% abs_mult_less
tff(fact_5296_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq2
tff(fact_5297_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq2
tff(fact_5298_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq2
tff(fact_5299_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq2
tff(fact_5300_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).

% abs_triangle_ineq3
tff(fact_5301_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).

% abs_triangle_ineq3
tff(fact_5302_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)))) ).

% abs_triangle_ineq3
tff(fact_5303_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).

% abs_triangle_ineq3
tff(fact_5304_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5305_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5306_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5307_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)))) ).

% abs_triangle_ineq2_sym
tff(fact_5308_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_5309_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_5310_abs__ge__minus__self,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).

% abs_ge_minus_self
tff(fact_5311_abs__ge__minus__self,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A))) ).

% abs_ge_minus_self
tff(fact_5312_abs__ge__minus__self,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A))) ).

% abs_ge_minus_self
tff(fact_5313_abs__ge__minus__self,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).

% abs_ge_minus_self
tff(fact_5314_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_le_iff
tff(fact_5315_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_le_iff
tff(fact_5316_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_le_iff
tff(fact_5317_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_le_iff
tff(fact_5318_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ).

% abs_le_D2
tff(fact_5319_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).

% abs_le_D2
tff(fact_5320_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ).

% abs_le_D2
tff(fact_5321_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).

% abs_le_D2
tff(fact_5322_abs__leI,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)) ) ) ).

% abs_leI
tff(fact_5323_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).

% abs_leI
tff(fact_5324_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)) ) ) ).

% abs_leI
tff(fact_5325_abs__leI,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).

% abs_leI
tff(fact_5326_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).

% abs_less_iff
tff(fact_5327_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).

% abs_less_iff
tff(fact_5328_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)) ) ) ).

% abs_less_iff
tff(fact_5329_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).

% abs_less_iff
tff(fact_5330_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5331_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5332_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),A)) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5333_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5334_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5335_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5336_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_5337_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_5338_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5339_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,Q4)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5340_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q4)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,Q4)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5341_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q4)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N2),numera6620942414471956472nteger(Q4)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_5342_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_5343_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_5344_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N2),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_5345_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) )
     => ~ ! [D5: 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),D5)) ) ).

% mod_eqE
tff(fact_5346_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D5: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D5)) ) ).

% mod_eqE
tff(fact_5347_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_5348_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_5349_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_5350_mod__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)) = N2 )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) = zero_zero_nat ) )
      & ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)) != N2 )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)) ) ) ) ).

% mod_Suc
tff(fact_5351_mod__induct,axiom,
    ! [P: fun_nat_bool,N2: nat,P3: nat,M: nat] :
      ( pp(aa_nat_bool(P,N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P3))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P3))
         => ( ! [N: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P3))
               => ( pp(aa_nat_bool(P,N))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P3))) ) )
           => pp(aa_nat_bool(P,M)) ) ) ) ) ).

% mod_induct
tff(fact_5352_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M: nat,N2: nat] :
      ( ! [M5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M5),zero_zero_nat))
     => ( ! [M5: nat,N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M5),N)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M5),N)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N2)) ) ) ).

% gcd_nat_induct
tff(fact_5353_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),N2)) ) ).

% mod_less_divisor
tff(fact_5354_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N2))),N2)) ).

% mod_Suc_le_divisor
tff(fact_5355_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q2: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q2) ) ).

% mod_eq_0D
tff(fact_5356_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) ) ) ).

% mod_geq
tff(fact_5357_mod__if,axiom,
    ! [M: nat,N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = M ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) ) ) ) ).

% mod_if
tff(fact_5358_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) ) ) ).

% le_mod_geq
tff(fact_5359_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N2) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_5360_dbl__def,axiom,
    ! [X: real] : aa_real_real(neg_numeral_dbl_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X) ).

% dbl_def
tff(fact_5361_dbl__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_numeral_dbl_rat,X) = aa_rat_rat(plus_plus_rat(X),X) ).

% dbl_def
tff(fact_5362_dbl__def,axiom,
    ! [X: int] : aa_int_int(neg_numeral_dbl_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X) ).

% dbl_def
tff(fact_5363_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E)) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5364_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E)) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5365_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
          | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_5366_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
        & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
          | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_5367_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
        & ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
          | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_5368_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_5369_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5370_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5371_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5372_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_5373_abs__minus__le__zero,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real)) ).

% abs_minus_le_zero
tff(fact_5374_abs__minus__le__zero,axiom,
    ! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).

% abs_minus_le_zero
tff(fact_5375_abs__minus__le__zero,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat)) ).

% abs_minus_le_zero
tff(fact_5376_abs__minus__le__zero,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).

% abs_minus_le_zero
tff(fact_5377_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5378_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5379_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5380_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5381_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5382_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5383_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5384_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5385_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X),Y)) ) ) ).

% abs_div_pos
tff(fact_5386_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).

% abs_div_pos
tff(fact_5387_zero__le__power__abs,axiom,
    ! [A: code_integer,N2: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N2))) ).

% zero_le_power_abs
tff(fact_5388_zero__le__power__abs,axiom,
    ! [A: real,N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N2))) ).

% zero_le_power_abs
tff(fact_5389_zero__le__power__abs,axiom,
    ! [A: rat,N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),N2))) ).

% zero_le_power_abs
tff(fact_5390_zero__le__power__abs,axiom,
    ! [A: int,N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N2))) ).

% zero_le_power_abs
tff(fact_5391_abs__if__raw,axiom,
    ! [X2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X2) = aa_int_int(uminus_uminus_int,X2) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),zero_zero_int))
       => ( aa_int_int(abs_abs_int,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_5392_abs__if__raw,axiom,
    ! [X2: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X2) = aa_real_real(uminus_uminus_real,X2) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),zero_zero_real))
       => ( aa_real_real(abs_abs_real,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_5393_abs__if__raw,axiom,
    ! [X2: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X2) = aa_rat_rat(uminus_uminus_rat,X2) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_5394_abs__if__raw,axiom,
    ! [X2: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X2) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(X2) = X2 ) ) ) ).

% abs_if_raw
tff(fact_5395_abs__of__neg,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_5396_abs__of__neg,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_5397_abs__of__neg,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_5398_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_5399_abs__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
       => ( aa_int_int(abs_abs_int,A) = A ) ) ) ).

% abs_if
tff(fact_5400_abs__if,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_if
tff(fact_5401_abs__if,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_if
tff(fact_5402_abs__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(A) = A ) ) ) ).

% abs_if
tff(fact_5403_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5404_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5405_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5406_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_le_iff
tff(fact_5407_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).

% abs_triangle_ineq4
tff(fact_5408_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).

% abs_triangle_ineq4
tff(fact_5409_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))) ).

% abs_triangle_ineq4
tff(fact_5410_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).

% abs_triangle_ineq4
tff(fact_5411_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,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_5412_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),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_5413_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D))))) ).

% abs_diff_triangle_ineq
tff(fact_5414_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),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_5415_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5416_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R2))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),X))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5417_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R2))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),X))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(plus_plus_rat(A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5418_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2))) ) ) ).

% abs_diff_less_iff
tff(fact_5419_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5420_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5421_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
       => ( aa_int_int(modulo_modulo_int(A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5422_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5423_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5424_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B))) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5425_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5426_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5427_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N2),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5428_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_5429_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_5430_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(N2),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_5431_cong__exp__iff__simps_I6_J,axiom,
    ! [Q4: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(6)
tff(fact_5432_cong__exp__iff__simps_I6_J,axiom,
    ! [Q4: num,N2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(6)
tff(fact_5433_cong__exp__iff__simps_I6_J,axiom,
    ! [Q4: num,N2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(6)
tff(fact_5434_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q4: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(8)
tff(fact_5435_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q4: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(8)
tff(fact_5436_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q4: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(8)
tff(fact_5437_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_5438_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_5439_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_5440_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_5441_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_5442_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_5443_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_5444_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_5445_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_5446_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_5447_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_5448_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_5449_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_5450_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_5451_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_5452_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_5453_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_5454_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_5455_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_5456_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_5457_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_5458_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_5459_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_5460_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_5461_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_5462_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_5463_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_5464_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_5465_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_5466_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_5467_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_5468_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_5469_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_5470_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_5471_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_5472_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_5473_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)),N2)) ) ).

% mod_le_divisor
tff(fact_5474_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N2) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N2) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N2))) ) ) ) ) ).

% div_less_mono
tff(fact_5475_abs__real__def,axiom,
    ! [A: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
       => ( aa_real_real(abs_abs_real,A) = A ) ) ) ).

% abs_real_def
tff(fact_5476_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ~ ! [S2: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_5477_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ~ ! [S2: nat] : N2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_5478_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ? [Q2: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_5479_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L) )
       => eucl_rel_int(K,L,product_Pair_int_int(Q4,zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5480_div__mod__decomp,axiom,
    ! [A2: nat,N2: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N2)),N2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N2)) ).

% div_mod_decomp
tff(fact_5481_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),Q4))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)) ).

% mod_mult2_eq
tff(fact_5482_finite__abs__int__segment,axiom,
    ! [A: real] : finite_finite_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_bu,A))) ).

% finite_abs_int_segment
tff(fact_5483_finite__abs__int__segment,axiom,
    ! [A: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_bv,A))) ).

% finite_abs_int_segment
tff(fact_5484_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X)))) ).

% abs_add_one_gt_zero
tff(fact_5485_abs__add__one__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,X)))) ).

% abs_add_one_gt_zero
tff(fact_5486_abs__add__one__gt__zero,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,X)))) ).

% abs_add_one_gt_zero
tff(fact_5487_abs__add__one__gt__zero,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X)))) ).

% abs_add_one_gt_zero
tff(fact_5488_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( ( X != zero_z3403309356797280102nteger )
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),abs_abs_Code_integer(X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5489_Ints__nonzero__abs__ge1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( ( X != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5490_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( ( X != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5491_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( pp(aa_set_int_bool(member_int(X),ring_1_Ints_int))
     => ( ( X != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X))) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5492_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_5493_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N2)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_5494_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] : aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N2)))),aa_int_int(modulo_modulo_int(A),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_5495_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer))
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5496_Ints__nonzero__abs__less1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
       => ( X = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5497_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat))
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5498_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( pp(aa_set_int_bool(member_int(X),ring_1_Ints_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int))
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5499_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y,ring_11222124179247155820nteger)
       => ( ( X = Y )
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,Y))),one_one_Code_integer)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5500_Ints__eq__abs__less1,axiom,
    ! [X: real,Y: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( member_real(Y,ring_1_Ints_real)
       => ( ( X = Y )
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y))),one_one_real)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5501_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( member_rat(Y,ring_1_Ints_rat)
       => ( ( X = Y )
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),Y))),one_one_rat)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5502_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),ring_1_Ints_int))
     => ( pp(aa_set_int_bool(member_int(Y),ring_1_Ints_int))
       => ( ( X = Y )
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y))),one_one_int)) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5503_split__mod,axiom,
    ! [P: fun_nat_bool,M: nat,N2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)))
    <=> ( ( ( N2 = zero_zero_nat )
         => pp(aa_nat_bool(P,M)) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N2))
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),I3)),J3) )
               => pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_5504_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_5505_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5506_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5507_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),aa_rat_rat(abs_abs_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5508_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_le_square_iff
tff(fact_5509_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_5510_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_5511_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,X) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_5512_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_5513_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5514_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5515_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5516_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_5517_abs__sqrt__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
      ( ! [X4: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X4))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X4),aa_nat_Code_integer(power_8256067586552552935nteger(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5518_abs__sqrt__wlog,axiom,
    ! [P: fun_re1255401033344146673l_bool,X: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X4),aa_nat_real(power_power_real(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X)),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5519_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_bool,X: rat] :
      ( ! [X4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X4))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X4),aa_nat_rat(power_power_rat(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(P,aa_rat_rat(abs_abs_rat,X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5520_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_bool,X: int] :
      ( ! [X4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,X4),aa_nat_int(power_power_int(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_sqrt_wlog
tff(fact_5521_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5522_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5523_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5524_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y)) ) ) ).

% power2_le_iff_abs_le
tff(fact_5525_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_le_1
tff(fact_5526_abs__square__le__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_le_1
tff(fact_5527_abs__square__le__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_le_1
tff(fact_5528_abs__square__le__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_le_1
tff(fact_5529_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).

% abs_square_less_1
tff(fact_5530_abs__square__less__1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).

% abs_square_less_1
tff(fact_5531_abs__square__less__1,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)) ) ).

% abs_square_less_1
tff(fact_5532_abs__square__less__1,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).

% abs_square_less_1
tff(fact_5533_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5534_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5535_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5536_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_5537_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_5538_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_5539_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ).

% div_exp_mod_exp_eq
tff(fact_5540_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(divide_divide_int(aa_int_int(modulo_modulo_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% div_exp_mod_exp_eq
tff(fact_5541_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N2)) ).

% div_exp_mod_exp_eq
tff(fact_5542_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( eucl_rel_int(A,B,product_Pair_int_int(Q4,R2))
     => ( ( B != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B,product_Pair_int_int(aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),aa_int_int(uminus_uminus_int,Q4)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q4)),one_one_int)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int),zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_5543_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N2)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N2)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N2))) ) ) ).

% verit_le_mono_div
tff(fact_5544_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5545_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(divide_divide_int(A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5546_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5547_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = 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,N2),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5548_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) = 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,N2),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5549_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5550_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q4,R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q4)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5551_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
       => ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = modulo364778990260209775nteger(X,M) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5552_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5553_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(modulo_modulo_int(X),M) )
          | ( aa_int_int(modulo_modulo_int(X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(X),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5554_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5555_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5556_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(modulo_modulo_int(A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5557_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N2)),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,N2),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_5558_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N2),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(N2,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5559_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N2)),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(N2),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5560_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N2),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(N2,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5561_set__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N2)),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(N2),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5562_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N2)),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,N2),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_5563_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_5564_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5565_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5566_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(divide_divide_int(A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5567_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( eucl_rel_int(A,B,product_Pair_int_int(Q4,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B),product_Pair_int_int(Q4,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5568_suminf__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => ( suminf_real(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_5569_suminf__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_5570_suminf__zero,axiom,
    suminf_real(aTP_Lamm_bw) = zero_zero_real ).

% suminf_zero
tff(fact_5571_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_ap) = zero_zero_nat ).

% suminf_zero
tff(fact_5572_suminf__zero,axiom,
    suminf_int(aTP_Lamm_bx) = zero_zero_int ).

% suminf_zero
tff(fact_5573_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval
tff(fact_5574_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => topolo6980174941875973593q_real(aTP_Lamm_by(X)) ) ).

% monoseq_arctan_series
tff(fact_5575_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(arctan,X) = suminf_real(aTP_Lamm_bz(X)) ) ) ).

% arctan_series
tff(fact_5576_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ca) ).

% pi_series
tff(fact_5577_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [Y4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y4))),D5))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),B)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_5578_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(arctan,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_5579_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_5580_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_5581_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_5582_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z2)),one_one_int))
    <=> ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_5583_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% arctan_less_zero_iff
tff(fact_5584_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% zero_less_arctan_iff
tff(fact_5585_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% arctan_le_zero_iff
tff(fact_5586_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_arctan_iff
tff(fact_5587_zmod__numeral__Bit0,axiom,
    ! [V: num,W2: 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,W2))) = 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,W2))) ).

% zmod_numeral_Bit0
tff(fact_5588_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L))) ) ).

% abs_mod_less
tff(fact_5589_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_5590_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% arctan_less_iff
tff(fact_5591_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).

% arctan_monotone
tff(fact_5592_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(M),K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_5593_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_5594_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L))) ) ).

% neg_mod_bound
tff(fact_5595_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_5596_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(modulo_modulo_int(K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_5597_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
    <=> ? [Q6: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q6) ) ).

% zmod_eq_0_iff
tff(fact_5598_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(modulo_modulo_int(M),D) = zero_zero_int )
     => ? [Q2: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q2) ) ).

% zmod_eq_0D
tff(fact_5599_abs__zmult__eq__1,axiom,
    ! [M: int,N2: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N2)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_5600_pi__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).

% pi_gt_zero
tff(fact_5601_pi__not__less__zero,axiom,
    ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),zero_zero_real)) ).

% pi_not_less_zero
tff(fact_5602_pi__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).

% pi_ge_zero
tff(fact_5603_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_5604_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,S3))
    <=> ! [M6: int] :
        ? [N4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M6),aa_int_int(abs_abs_int,N4)))
          & pp(aa_set_int_bool(member_int(N4),S3)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_5605_mod__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,product_Pair_int_int(Q4,R2))
     => ( aa_int_int(modulo_modulo_int(K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_5606_arctan__ubound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% arctan_ubound
tff(fact_5607_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% arctan_bounded
tff(fact_5608_arctan__lbound,axiom,
    ! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))) ).

% arctan_lbound
tff(fact_5609_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_5610_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int)) ) ).

% neg_mod_sign
tff(fact_5611_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(modulo_modulo_int(A),B)),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(modulo_modulo_int(A),B))) ) ) ).

% neg_mod_conj
tff(fact_5612_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A),B)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(modulo_modulo_int(A),B)),B)) ) ) ).

% pos_mod_conj
tff(fact_5613_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_5614_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
       => ( aa_int_int(abs_abs_int,I) = I ) ) ) ).

% zabs_def
tff(fact_5615_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N2))
       => ( ( aa_int_int(modulo_modulo_int(A2),N2) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B2),N2) = zero_zero_int )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(divide_divide_int(A2),N2)),aa_int_int(divide_divide_int(B2),N2))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_5616_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(A),B)),B) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_5617_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
      & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(modulo_modulo_int(A),B)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_5618_div__mod__decomp__int,axiom,
    ! [A2: int,N2: int] : A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(divide_divide_int(A2),N2)),N2)),aa_int_int(modulo_modulo_int(A2),N2)) ).

% div_mod_decomp_int
tff(fact_5619_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,product_Pair_int_int(aa_int_int(divide_divide_int(K),L),aa_int_int(modulo_modulo_int(K),L))) ).

% eucl_rel_int
tff(fact_5620_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_5621_pi__less__4,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) ).

% pi_less_4
tff(fact_5622_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_5623_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: 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),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),B))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_5624_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: 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),Q4)),R2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R2))
         => ( aa_int_int(modulo_modulo_int(A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_5625_split__zmod,axiom,
    ! [P: fun_int_bool,N2: int,K: int] :
      ( pp(aa_int_bool(P,aa_int_int(modulo_modulo_int(N2),K)))
    <=> ( ( ( K = zero_zero_int )
         => pp(aa_int_bool(P,N2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
         => ! [I3: int,J3: int] :
              ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
                & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => pp(aa_int_bool(P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_5626_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_5627_zmod__minus1,axiom,
    ! [B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5628_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_5629_pi__half__less__two,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_less_two
tff(fact_5630_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( aa_int_int(modulo_modulo_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(modulo_modulo_int(aa_int_int(divide_divide_int(A),B)),C))),aa_int_int(modulo_modulo_int(A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_5631_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_5632_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
         => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B)) ) )
        & ( ( aa_int_int(modulo_modulo_int(A),B) != zero_zero_int )
         => ( aa_int_int(divide_divide_int(A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(A),B))),one_one_int) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_5633_monoseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => topolo6980174941875973593q_real(power_power_real(X)) ) ) ).

% monoseq_realpow
tff(fact_5634_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N2)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N2)))
           => ? [I2: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N2))
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_5635_decr__lemma,axiom,
    ! [D: int,X: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z2))),one_one_int)),D))),Z2)) ) ).

% decr_lemma
tff(fact_5636_incr__lemma,axiom,
    ! [D: int,Z2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z2))),one_one_int)),D)))) ) ).

% incr_lemma
tff(fact_5637_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A2),N2)),aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(B2),N2)),zero_zero_int),one_one_int),zero_zero_int))),aa_int_int(divide_divide_int(B2),N2))) ) ) ).

% verit_le_mono_div_int
tff(fact_5638_pi__half__gt__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_gt_zero
tff(fact_5639_pi__half__ge__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% pi_half_ge_zero
tff(fact_5640_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(N2),K)),aa_int_int(modulo_modulo_int(N2),K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
              & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_5641_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_bool,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(N2),K)),aa_int_int(modulo_modulo_int(N2),K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
              & ( N2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => pp(aa_int_bool(aa_int_fun_int_bool(P,I3),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_5642_m2pi__less__pi,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_5643_nat__ivt__aux,axiom,
    ! [N2: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N2)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N2))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_5644_minus__pi__half__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real)) ).

% minus_pi_half_less_zero
tff(fact_5645_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_5646_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N2)))
         => ? [I2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N2))
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_5647_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).

% arctan_add
tff(fact_5648_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
     => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_5649_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_5650_sin__cos__npi,axiom,
    ! [N2: 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))),N2)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2) ).

% sin_cos_npi
tff(fact_5651_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N2),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(N2,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5652_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N2)),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(N2),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5653_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N2)),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,N2),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_5654_signed__take__bit__rec,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( ( N2 = zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N2,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) )
      & ( ( N2 != zero_zero_nat )
       => ( bit_ri6519982836138164636nteger(N2,A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ) ) ) ).

% signed_take_bit_rec
tff(fact_5655_signed__take__bit__rec,axiom,
    ! [N2: nat,A: int] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N2),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)))) ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_int_int(bit_ri631733984087533419it_int(N2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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_5656_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => summable_real(aTP_Lamm_bz(X)) ) ).

% summable_arctan_series
tff(fact_5657_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys2: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys2))))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys2),N2) = product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_5658_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2),N2) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_5659_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys2),N2) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_5660_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys2))))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys2),N2) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_5661_product__nth,axiom,
    ! [N2: nat,Xs: list_bool,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys2),N2) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_5662_product__nth,axiom,
    ! [N2: nat,Xs: list_bool,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr2856333732135533678l_bool(product_bool_bool(Xs,Ys2),N2) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_5663_product__nth,axiom,
    ! [N2: nat,Xs: list_bool,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_nat(Ys2))))
     => ( nth_Pr2186422209488468047ol_nat(product_bool_nat(Xs,Ys2),N2) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_5664_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_VEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys2))))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys2),N2) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_5665_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_bool(Ys2))))
     => ( nth_Pr396318242284072693t_bool(product_nat_bool(Xs,Ys2),N2) = produc3913204875133826030t_bool(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_bool(Ys2))),aa_nat_bool(nth_bool(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_bool(Ys2)))) ) ) ).

% product_nth
tff(fact_5666_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys2: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys2))))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys2),N2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),size_size_list_nat(Ys2)))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_5667_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5668_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_5669_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_5670_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_5671_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N2),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% flip_bit_negative_int_iff
tff(fact_5672_signed__take__bit__of__0,axiom,
    ! [N2: nat] : aa_int_int(bit_ri631733984087533419it_int(N2),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_5673_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_5674_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(tanh_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_5675_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% tanh_real_less_iff
tff(fact_5676_summable__zero,axiom,
    summable_real(aTP_Lamm_bw) ).

% summable_zero
tff(fact_5677_summable__zero,axiom,
    summable_nat(aTP_Lamm_ap) ).

% summable_zero
tff(fact_5678_summable__zero,axiom,
    summable_int(aTP_Lamm_bx) ).

% summable_zero
tff(fact_5679_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cb(I),F)) ).

% summable_single
tff(fact_5680_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cc(I),F)) ).

% summable_single
tff(fact_5681_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_cd(I),F)) ).

% summable_single
tff(fact_5682_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_5683_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_5684_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_5685_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_5686_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] : bit_ri6519982836138164636nteger(N2,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_5687_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] : aa_int_int(bit_ri631733984087533419it_int(N2),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_5688_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_5689_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_5690_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% tanh_real_pos_iff
tff(fact_5691_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% tanh_real_neg_iff
tff(fact_5692_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% tanh_real_nonpos_iff
tff(fact_5693_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_5694_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cf(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_5695_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cg(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_5696_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ch(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_5697_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ci(P),F)) ) ).

% summable_If_finite
tff(fact_5698_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cj(P),F)) ) ).

% summable_If_finite
tff(fact_5699_summable__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ck(P),F)) ) ).

% summable_If_finite
tff(fact_5700_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cl(A2),F)) ) ).

% summable_If_finite_set
tff(fact_5701_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cm(A2),F)) ) ).

% summable_If_finite_set
tff(fact_5702_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_cn(A2),F)) ) ).

% summable_If_finite_set
tff(fact_5703_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_5704_sin__cos__squared__add3,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_5705_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_5706_sin__npi2,axiom,
    ! [N2: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N2))) = zero_zero_real ).

% sin_npi2
tff(fact_5707_sin__npi,axiom,
    ! [N2: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_5708_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(power_power_real(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_5709_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)) ) ).

% summable_geometric_iff
tff(fact_5710_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5711_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_5712_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_5713_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_5714_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_5715_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_5716_sin__cos__squared__add,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_5717_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_5718_sin__cos__squared__add2,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_5719_sin__2npi,axiom,
    ! [N2: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N2))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_5720_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) = one_one_complex )
     => ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_5721_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
     => ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_5722_sin__add,axiom,
    ! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_5723_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N5: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5724_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N5: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5725_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5726_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5727_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aTP_Lamm_co(C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_5728_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cp(F),G)) ) ) ).

% summable_add
tff(fact_5729_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cq(F),G)) ) ) ).

% summable_add
tff(fact_5730_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_cr(F),G)) ) ) ).

% summable_add
tff(fact_5731_cos__add,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_5732_cos__diff,axiom,
    ! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_5733_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_cs(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_5734_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_5735_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_5736_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_5737_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_5738_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z2)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_cu(F),Z2)) ) ) ).

% powser_insidea
tff(fact_5739_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_cw(F),Z2)) ) ) ).

% powser_insidea
tff(fact_5740_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_5741_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N)))
     => ( summable_real(F)
       => ( summable_real(G)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),suminf_real(G))) ) ) ) ).

% suminf_le
tff(fact_5742_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N)))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G))) ) ) ) ).

% suminf_le
tff(fact_5743_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N)))
     => ( summable_int(F)
       => ( summable_int(G)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),suminf_int(G))) ) ) ) ).

% suminf_le
tff(fact_5744_summable__finite,axiom,
    ! [N5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_5745_summable__finite,axiom,
    ! [N5: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_5746_summable__finite,axiom,
    ! [N5: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_5747_sin__x__le__x,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),X)) ) ).

% sin_x_le_x
tff(fact_5748_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_cf(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_5749_summable__zero__power,axiom,
    summable_real(power_power_real(zero_zero_real)) ).

% summable_zero_power
tff(fact_5750_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_5751_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_5752_cos__arctan__not__zero,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,X)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_5753_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_cp(F),G)) ) ) ) ).

% suminf_add
tff(fact_5754_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_cq(F),G)) ) ) ) ).

% suminf_add
tff(fact_5755_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_cr(F),G)) ) ) ) ).

% suminf_add
tff(fact_5756_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),one_one_real)) ).

% tanh_real_lt_1
tff(fact_5757_cos__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_5758_cos__squared__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_5759_sin__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_5760_sin__squared__eq,axiom,
    ! [X: real] : aa_nat_real(power_power_real(aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_5761_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5762_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5763_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(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_5764_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_nonneg
tff(fact_5765_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_nonneg
tff(fact_5766_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_nonneg
tff(fact_5767_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).

% suminf_pos
tff(fact_5768_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).

% suminf_pos
tff(fact_5769_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).

% suminf_pos
tff(fact_5770_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero
tff(fact_5771_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_5772_sin__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_ge_zero
tff(fact_5773_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_bs(F)) ).

% summable_zero_power'
tff(fact_5774_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_bt(F)) ).

% summable_zero_power'
tff(fact_5775_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_cx(F)) ).

% summable_zero_power'
tff(fact_5776_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_bs(F)) ).

% summable_0_powser
tff(fact_5777_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_bt(F)) ).

% summable_0_powser
tff(fact_5778_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_5779_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_5780_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( ( aa_real_real(cos_real,X) = aa_real_real(cos_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_5781_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),pi)) = zero_zero_real )
    <=> member_real(X,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_5782_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cy(F),Z2))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2)) ) ).

% summable_powser_split_head
tff(fact_5783_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_cz(F),Z2))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2)) ) ).

% summable_powser_split_head
tff(fact_5784_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cy(F),Z2)) ) ).

% powser_split_head(3)
tff(fact_5785_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_cz(F),Z2)) ) ).

% powser_split_head(3)
tff(fact_5786_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,M: nat,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_da(F),M),Z2))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_5787_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,M: nat,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_db(F),M),Z2))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_5788_sin__times__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z2)) = 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,W2),Z2))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_5789_sin__times__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W2),Z2))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_5790_sin__times__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W2)),aa_real_real(cos_real,Z2)) = 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,W2),Z2))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_5791_sin__times__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W2),Z2))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_5792_cos__times__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W2)),aa_real_real(sin_real,Z2)) = 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,W2),Z2))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_5793_cos__times__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W2),Z2))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W2),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_5794_sin__plus__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_5795_sin__plus__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z2)) = 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(W2),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_5796_sin__diff__sin,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W2)),aa_real_real(sin_real,Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_5797_sin__diff__sin,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W2)),aa_complex_complex(sin_complex,Z2)) = 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(W2),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W2),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_5798_cos__diff__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),W2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_5799_cos__diff__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z2)) = 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(W2),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z2),W2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_5800_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_dc(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_5801_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_dd(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_5802_cos__double__sin,axiom,
    ! [W2: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W2)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_5803_cos__double__sin,axiom,
    ! [W2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(aa_real_real(sin_real,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_5804_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).

% suminf_pos2
tff(fact_5805_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).

% suminf_pos2
tff(fact_5806_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).

% suminf_pos2
tff(fact_5807_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
        <=> ? [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_5808_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
        <=> ? [I3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_5809_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
        <=> ? [I3: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I3))) ) ) ) ).

% suminf_pos_iff
tff(fact_5810_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X))) ).

% tanh_real_gt_neg1
tff(fact_5811_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_5812_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z2)),aa_real_real(real_V7735802525324610683m_real,X)))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2)) ) ) ).

% powser_inside
tff(fact_5813_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(real_V1022390504157884413omplex,X)))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2)) ) ) ).

% powser_inside
tff(fact_5814_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_5815_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_5816_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
       => ( ( aa_real_real(sin_real,X) = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_5817_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real))
     => summable_real(power_power_real(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_5818_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real))
     => summable_complex(power_power_complex(X)) ) ).

% complete_algebra_summable_geometric
tff(fact_5819_summable__geometric,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => summable_real(power_power_real(C)) ) ).

% summable_geometric
tff(fact_5820_summable__geometric,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => summable_complex(power_power_complex(C)) ) ).

% summable_geometric
tff(fact_5821_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),pi))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_5822_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_cs(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_5823_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_5824_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),pi))
            & ( X = aa_real_real(cos_real,T2) )
            & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_5825_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_5826_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_5827_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_5828_cos__two__less__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_less_zero
tff(fact_5829_cos__two__le__zero,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real)) ).

% cos_two_le_zero
tff(fact_5830_cos__is__zero,axiom,
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(cos_real,X4) = zero_zero_real )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
tff(fact_5831_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_5832_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ? [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),pi))
            & ( aa_real_real(cos_real,X4) = Y )
            & ! [Y4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),pi))
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_5833_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N2),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ).

% signed_take_bit_int_less_exp
tff(fact_5834_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
              & ( X = aa_real_real(cos_real,T2) )
              & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_5835_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
          & ( X = aa_real_real(cos_real,T2) )
          & ( Y = aa_real_real(sin_real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_5836_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( ( X = aa_real_real(cos_real,T2) )
               => ( Y != aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_5837_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2)) = 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_cy(F),Z2))),Z2)) ) ) ).

% powser_split_head(1)
tff(fact_5838_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2)) = 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_cz(F),Z2))),Z2)) ) ) ).

% powser_split_head(1)
tff(fact_5839_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_cy(F),Z2))),Z2) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(F),Z2))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_5840_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_cz(F),Z2))),Z2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_5841_sin__integer__2pi,axiom,
    ! [N2: real] :
      ( member_real(N2,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),N2)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_5842_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_real(F)
       => ? [N8: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),N6)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_5843_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
     => ( summable_complex(F)
       => ? [N8: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_de(F),N6)))),R2)) ) ) ) ).

% suminf_exist_split
tff(fact_5844_sin__pi__divide__n__ge__0,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N2))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_5845_summable__power__series,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real))
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),one_one_real))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(F),Z2)) ) ) ) ) ).

% summable_power_series
tff(fact_5846_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M7: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),R0))
       => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(power_power_real(R0),N))),M7))
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_df(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_5847_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N2),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_5848_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_5849_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5850_cos__times__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z2)) = 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,W2),Z2))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_5851_cos__times__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W2),Z2))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W2),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_5852_cos__plus__cos,axiom,
    ! [W2: real,Z2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W2)),aa_real_real(cos_real,Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W2),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_5853_cos__plus__cos,axiom,
    ! [W2: complex,Z2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W2)),aa_complex_complex(cos_complex,Z2)) = 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(W2),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W2),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_5854_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,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_5855_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
     => ( ! [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),N))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5856_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).

% sin_gt_zero2
tff(fact_5857_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_lt_zero
tff(fact_5858_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).

% cos_double_less_one
tff(fact_5859_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero
tff(fact_5860_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N2),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N2))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5861_cos__double__cos,axiom,
    ! [W2: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_5862_cos__double__cos,axiom,
    ! [W2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(power_power_real(aa_real_real(cos_real,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_5863_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))
       => ( aa_int_int(bit_ri631733984087533419it_int(N2),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5864_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N2),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5865_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_le_zero
tff(fact_5866_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).

% sin_less_zero
tff(fact_5867_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_5868_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_5869_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_5870_cos__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).

% cos_ge_zero
tff(fact_5871_sin__pi__divide__n__gt__0,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N2))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_5872_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N2)))),aa_int_int(bit_ri631733984087533419it_int(N2),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5873_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N2),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(N2,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5874_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),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(N2),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5875_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,X))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_5876_tan__double,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_5877_sum__gp,axiom,
    ! [N2: nat,M: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = zero_zero_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
       => ( ( ( X = one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = 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,N2),one_one_nat)),M)) ) )
          & ( ( X != one_one_complex )
           => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N2)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5878_sum__gp,axiom,
    ! [N2: nat,M: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
       => ( ( ( X = one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = 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,N2),one_one_nat)),M)) ) )
          & ( ( X != one_one_rat )
           => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5879_sum__gp,axiom,
    ! [N2: nat,M: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
       => ( ( ( X = one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = 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,N2),one_one_nat)),M)) ) )
          & ( ( X != one_one_real )
           => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ) ) ).

% sum_gp
tff(fact_5880_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dg(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_dg(I5),Y)))
       => finite_finite_real(collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_dh(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5881_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_di(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_di(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_dj(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5882_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_complex,Y: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dk(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_dk(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_dl(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5883_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dm(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_dm(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_dn(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5884_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex3741802288643809319omplex,Y: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun7434105802671982559t_bool(aTP_Lamm_do(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun7434105802671982559t_bool(aTP_Lamm_do(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun7434105802671982559t_bool(aa_fun6445447259835133315t_bool(aTP_Lamm_dp(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5885_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dq(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_dq(I5),Y)))
       => finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_dr(I5),X),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5886_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ds(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ds(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_dt(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5887_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_du(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_du(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_dv(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5888_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dw(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_dw(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_dx(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5889_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex1451620480498224165t_real,Y: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_dy(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_dy(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_dz(I5),X),Y)))) ) ) ).

% prod.finite_Collect_op
tff(fact_5890_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_ea(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_ea(I5),Y)))
       => finite_finite_real(collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_eb(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5891_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ec(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_ec(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_ed(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5892_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ee(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_ee(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ef(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5893_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_eg(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_eg(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_eh(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5894_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex1451620480498224165t_real,Y: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_ei(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aTP_Lamm_ei(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_ej(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5895_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_ek(I5),X)))
     => ( finite_finite_real(collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_ek(I5),Y)))
       => finite_finite_real(collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_el(I5),X),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5896_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_em(I5),X))))
     => ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_em(I5),Y))))
       => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_en(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5897_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: fun_int_rat,Y: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_eo(I5),X))))
     => ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aTP_Lamm_eo(I5),Y))))
       => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_ep(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5898_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: fun_complex_rat,Y: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_eq(I5),X))))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aTP_Lamm_eq(I5),Y))))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2759627087638585025x_bool(aa_fun1496239077807485399x_bool(aTP_Lamm_er(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5899_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X: fun_Ex4152069839811072081at_rat,Y: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun5314330785533781769t_bool(aTP_Lamm_es(I5),X))))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun5314330785533781769t_bool(aTP_Lamm_es(I5),Y))))
       => pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,collec4429806609662206161d_enat(aa_fun5314330785533781769t_bool(aa_fun4769541129632704515t_bool(aTP_Lamm_et(I5),X),Y)))) ) ) ).

% sum.finite_Collect_op
tff(fact_5900_geometric__deriv__sums,axiom,
    ! [Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z2)),one_one_real))
     => pp(aa_real_bool(sums_real(aTP_Lamm_eu(Z2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_5901_geometric__deriv__sums,axiom,
    ! [Z2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),one_one_real))
     => pp(aa_complex_bool(sums_complex(aTP_Lamm_ev(Z2)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% geometric_deriv_sums
tff(fact_5902_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_5903_dvd__0__right,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),zero_zero_real)) ).

% dvd_0_right
tff(fact_5904_dvd__0__right,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),zero_zero_rat)) ).

% dvd_0_right
tff(fact_5905_dvd__0__right,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% dvd_0_right
tff(fact_5906_dvd__0__right,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),zero_zero_int)) ).

% dvd_0_right
tff(fact_5907_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_5908_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_5909_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_5910_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_5911_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_5912_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_5913_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_5914_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_right_iff
tff(fact_5915_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_5916_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_5917_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_5918_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_triv_left_iff
tff(fact_5919_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_5920_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(B),A)),aa_int_int(divide_divide_int(C),A)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ) ).

% div_dvd_div
tff(fact_5921_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% dvd_minus_iff
tff(fact_5922_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% dvd_minus_iff
tff(fact_5923_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_complex_complex(uminus1482373934393186551omplex,Y)))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).

% dvd_minus_iff
tff(fact_5924_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_rat_rat(uminus_uminus_rat,Y)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% dvd_minus_iff
tff(fact_5925_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% dvd_minus_iff
tff(fact_5926_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).

% minus_dvd_iff
tff(fact_5927_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).

% minus_dvd_iff
tff(fact_5928_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(uminus1482373934393186551omplex,X)),Y))
    <=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).

% minus_dvd_iff
tff(fact_5929_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,X)),Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).

% minus_dvd_iff
tff(fact_5930_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).

% minus_dvd_iff
tff(fact_5931_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% abs_dvd_iff
tff(fact_5932_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% abs_dvd_iff
tff(fact_5933_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,M)),K))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% abs_dvd_iff
tff(fact_5934_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% abs_dvd_iff
tff(fact_5935_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),aa_real_real(abs_abs_real,K)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).

% dvd_abs_iff
tff(fact_5936_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),aa_int_int(abs_abs_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).

% dvd_abs_iff
tff(fact_5937_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),aa_rat_rat(abs_abs_rat,K)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).

% dvd_abs_iff
tff(fact_5938_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).

% dvd_abs_iff
tff(fact_5939_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_5940_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_5941_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_5942_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_set_int_bool(member_int(aa_complex_int(F,X4)),ring_1_Ints_int)) )
     => pp(aa_set_int_bool(member_int(groups5690904116761175830ex_int(F,A2)),ring_1_Ints_int)) ) ).

% Ints_sum
tff(fact_5943_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_set_int_bool(member_int(aa_real_int(F,X4)),ring_1_Ints_int)) )
     => pp(aa_set_int_bool(member_int(groups1932886352136224148al_int(F,A2)),ring_1_Ints_int)) ) ).

% Ints_sum
tff(fact_5944_Ints__sum,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_int_bool(member_int(aa_nat_int(F,X4)),ring_1_Ints_int)) )
     => pp(aa_set_int_bool(member_int(groups3539618377306564664at_int(F,A2)),ring_1_Ints_int)) ) ).

% Ints_sum
tff(fact_5945_Ints__sum,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => member_real(aa_complex_real(F,X4),ring_1_Ints_real) )
     => member_real(groups5808333547571424918x_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_5946_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => member_real(aa_real_real(F,X4),ring_1_Ints_real) )
     => member_real(groups8097168146408367636l_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_5947_Ints__sum,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => member_real(aa_int_real(F,X4),ring_1_Ints_real) )
     => member_real(groups8778361861064173332t_real(F,A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_5948_Ints__sum,axiom,
    ! [A2: set_real,F: fun_real_complex] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_set_complex_bool(member_complex(aa_real_complex(F,X4)),ring_1_Ints_complex)) )
     => pp(aa_set_complex_bool(member_complex(groups5754745047067104278omplex(F,A2)),ring_1_Ints_complex)) ) ).

% Ints_sum
tff(fact_5949_Ints__sum,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_complex_bool(member_complex(aa_nat_complex(F,X4)),ring_1_Ints_complex)) )
     => pp(aa_set_complex_bool(member_complex(groups2073611262835488442omplex(F,A2)),ring_1_Ints_complex)) ) ).

% Ints_sum
tff(fact_5950_Ints__sum,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_set_complex_bool(member_complex(aa_int_complex(F,X4)),ring_1_Ints_complex)) )
     => pp(aa_set_complex_bool(member_complex(groups3049146728041665814omplex(F,A2)),ring_1_Ints_complex)) ) ).

% Ints_sum
tff(fact_5951_Ints__sum,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => member_real(aa_nat_real(F,X4),ring_1_Ints_real) )
     => member_real(aa_set_nat_real(groups6591440286371151544t_real(F),A2),ring_1_Ints_real) ) ).

% Ints_sum
tff(fact_5952_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ap),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5953_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bw),A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_5954_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_ew,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_5955_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_aq,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_5956_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_ex(F),A2) ).

% of_nat_sum
tff(fact_5957_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_ey(F),A2) ).

% of_nat_sum
tff(fact_5958_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups2073611262835488442omplex(aTP_Lamm_ez(F),A2) ).

% of_nat_sum
tff(fact_5959_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups2906978787729119204at_rat(aTP_Lamm_fa(F),A2) ).

% of_nat_sum
tff(fact_5960_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_fb(F),A2) ).

% of_nat_sum
tff(fact_5961_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fc(F)),A2) ).

% of_nat_sum
tff(fact_5962_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fd(F)),A2) ).

% of_nat_sum
tff(fact_5963_sums__zero,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_bw),zero_zero_real)) ).

% sums_zero
tff(fact_5964_sums__zero,axiom,
    pp(aa_nat_bool(sums_nat(aTP_Lamm_ap),zero_zero_nat)) ).

% sums_zero
tff(fact_5965_sums__zero,axiom,
    pp(aa_int_bool(sums_int(aTP_Lamm_bx),zero_zero_int)) ).

% sums_zero
tff(fact_5966_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5967_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5968_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_left
tff(fact_5969_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
    <=> ( ( C = zero_zero_real )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5970_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
    <=> ( ( C = zero_zero_rat )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5971_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
    <=> ( ( C = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_mult_cancel_right
tff(fact_5972_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5973_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5974_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5975_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5976_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5977_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C),A))))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5978_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5979_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5980_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5981_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),A)),B)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5982_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5983_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5984_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ).

% unit_prod
tff(fact_5985_unit__prod,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ).

% unit_prod
tff(fact_5986_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_5987_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_5988_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_5989_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_5990_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5991_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_5992_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5993_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_5994_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(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_5995_unit__div__1__unit,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(one_one_int),A)),one_one_int)) ) ).

% unit_div_1_unit
tff(fact_5996_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat)) ) ) ).

% unit_div
tff(fact_5997_unit__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),one_one_int)) ) ) ).

% unit_div
tff(fact_5998_sum_Oempty,axiom,
    ! [G: fun_Ex1451620480498224165t_real] : groups4148127829035722712t_real(G,bot_bo7653980558646680370d_enat) = zero_zero_real ).

% sum.empty
tff(fact_5999_sum_Oempty,axiom,
    ! [G: fun_Ex4152069839811072081at_rat] : groups1392844769737527556at_rat(G,bot_bo7653980558646680370d_enat) = zero_zero_rat ).

% sum.empty
tff(fact_6000_sum_Oempty,axiom,
    ! [G: fun_Ex7660298398204468809at_nat] : groups2027974829824023292at_nat(G,bot_bo7653980558646680370d_enat) = zero_zero_nat ).

% sum.empty
tff(fact_6001_sum_Oempty,axiom,
    ! [G: fun_Ex3482447378695272101at_int] : groups2025484359314973016at_int(G,bot_bo7653980558646680370d_enat) = zero_zero_int ).

% sum.empty
tff(fact_6002_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_6003_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_6004_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_6005_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_6006_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_6007_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_6008_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,F3))
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),F3))
           => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6009_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,F3))
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),F3))
           => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6010_sum__eq__0__iff,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,F3))
     => ( ( groups977919841031483927at_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),F3))
           => ( aa_Pro4243173770044397973at_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6011_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,F3))
     => ( ( groups2027974829824023292at_nat(F,F3) = zero_zero_nat )
      <=> ! [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),F3))
           => ( aa_Extended_enat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6012_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,F3))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),F3))
           => ( aa_nat_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6013_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_6014_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_6015_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4148127829035722712t_real(G,A2) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_6016_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2906978787729119204at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6017_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6018_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6019_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups1392844769737527556at_rat(G,A2) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6020_sum_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6021_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5693394587270226106ex_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6022_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups2027974829824023292at_nat(G,A2) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6023_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_6024_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_6025_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_6026_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_6027_dvd__1__left,axiom,
    ! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).

% dvd_1_left
tff(fact_6028_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_6029_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))
    <=> ( ( K = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_6030_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fe(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fe(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6031_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ff(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ff(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6032_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fg(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fg(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6033_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_fh(A),B),S3) = aa_Ext2541572767299617110t_real(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_fh(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta'
tff(fact_6034_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fi(A),B),S3) = aa_real_rat(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fi(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_6035_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fj(A),B),S3) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fj(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_6036_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fk(A),B),S3) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fk(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_6037_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fl(A),B),S3) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fl(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_6038_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_fm(A),B),S3) = aa_Extended_enat_rat(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_fm(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta'
tff(fact_6039_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_nat] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fn(A),B),S3) = aa_real_nat(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fn(A),B),S3) = zero_zero_nat ) ) ) ) ).

% sum.delta'
tff(fact_6040_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fo(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fo(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6041_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fp(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fp(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6042_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fq(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fq(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6043_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_fr(A),B),S3) = aa_Ext2541572767299617110t_real(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4148127829035722712t_real(aa_fun1749635019461158908t_real(aTP_Lamm_fr(A),B),S3) = zero_zero_real ) ) ) ) ).

% sum.delta
tff(fact_6044_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fs(A),B),S3) = aa_real_rat(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fs(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_6045_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ft(A),B),S3) = aa_nat_rat(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ft(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_6046_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fu(A),B),S3) = aa_int_rat(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fu(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_6047_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fv(A),B),S3) = aa_complex_rat(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fv(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_6048_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_fw(A),B),S3) = aa_Extended_enat_rat(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups1392844769737527556at_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_fw(A),B),S3) = zero_zero_rat ) ) ) ) ).

% sum.delta
tff(fact_6049_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_nat] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fx(A),B),S3) = aa_real_nat(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fx(A),B),S3) = zero_zero_nat ) ) ) ) ).

% sum.delta
tff(fact_6050_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dd(F)),A2))) ).

% sum_abs
tff(fact_6051_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_fy(F),A2))) ).

% sum_abs
tff(fact_6052_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6053_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6054_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6055_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6056_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_6057_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(B),N2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% pow_divides_pow_iff
tff(fact_6058_tan__npi,axiom,
    ! [N2: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_6059_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dd(F)),A2))) ).

% sum_abs_ge_zero
tff(fact_6060_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_fy(F),A2))) ).

% sum_abs_ge_zero
tff(fact_6061_even__add,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_6062_even__add,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% even_add
tff(fact_6063_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
    <=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_6064_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
    <=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).

% odd_add
tff(fact_6065_even__Suc,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N2)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) ).

% even_Suc
tff(fact_6066_even__Suc__Suc__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,N2))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) ) ).

% even_Suc_Suc_iff
tff(fact_6067_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( pp(aa_complex_bool(sums_complex(aTP_Lamm_bs(A)),X))
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_6068_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_bt(A)),X))
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_6069_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_6070_even__plus__one__iff,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).

% even_plus_one_iff
tff(fact_6071_even__diff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ).

% even_diff
tff(fact_6072_odd__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N2)),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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_6073_even__Suc__div__two,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_6074_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = zero_zero_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6075_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = zero_zero_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6076_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6077_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_6078_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)) ) ).

% set_decode_0
tff(fact_6079_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_bs(C),A2) = aa_nat_complex(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aTP_Lamm_bs(C),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power
tff(fact_6080_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_fz(C),A2) = aa_nat_rat(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aTP_Lamm_fz(C),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power
tff(fact_6081_sum__zero__power,axiom,
    ! [A2: set_nat,C: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bt(C)),A2) = aa_nat_real(C,zero_zero_nat) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_bt(C)),A2) = zero_zero_real ) ) ) ).

% sum_zero_power
tff(fact_6082_even__succ__div__2,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6083_even__succ__div__2,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6084_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_6085_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_6086_even__succ__div__two,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6087_even__succ__div__two,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6088_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6089_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6090_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6091_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% even_power
tff(fact_6092_even__power,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),N2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).

% even_power
tff(fact_6093_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6094_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6095_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6096_power__less__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(A),N2)),zero_zero_real))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% power_less_zero_eq
tff(fact_6097_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),zero_zero_rat))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% power_less_zero_eq
tff(fact_6098_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N2)),zero_zero_int))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% power_less_zero_eq
tff(fact_6099_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_6100_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_6101_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_6102_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_6103_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_6104_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_6105_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_6106_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_6107_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_6108_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_6109_odd__Suc__minus__one,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))) = N2 ) ) ).

% odd_Suc_minus_one
tff(fact_6110_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ) ) ).

% even_diff_nat
tff(fact_6111_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_rat,D: fun_nat_rat] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ga(C),D),A2) = aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ga(C),D),A2) = zero_zero_rat ) ) ) ).

% sum_zero_power'
tff(fact_6112_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_complex,D: fun_nat_complex] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gb(C),D),A2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gb(C),D),A2) = zero_zero_complex ) ) ) ).

% sum_zero_power'
tff(fact_6113_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: fun_nat_real,D: fun_nat_real] :
      ( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
          & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gc(C),D)),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
      & ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
            & pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gc(C),D)),A2) = zero_zero_real ) ) ) ).

% sum_zero_power'
tff(fact_6114_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_6115_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_6116_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),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))),N2)),one_one_nat)))
    <=> ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6117_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),one_one_int)))
    <=> ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6118_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6119_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6120_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6121_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( 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,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_6122_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6123_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6124_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2)))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W2)))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6125_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( 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))),N2)) = 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))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_6126_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( 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))),N2)) = aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ) ) ) ).

% even_succ_div_exp
tff(fact_6127_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( 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))),N2)) = 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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6128_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( 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))),N2)) = 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))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6129_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N2)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N2))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6130_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5))) ) ).

% sum_mono
tff(fact_6131_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5))) ) ).

% sum_mono
tff(fact_6132_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5))) ) ).

% sum_mono
tff(fact_6133_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5))) ) ).

% sum_mono
tff(fact_6134_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5))) ) ).

% sum_mono
tff(fact_6135_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5))) ) ).

% sum_mono
tff(fact_6136_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),K5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5))) ) ).

% sum_mono
tff(fact_6137_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5))) ) ).

% sum_mono
tff(fact_6138_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( member_real(I2,K5)
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5))) ) ).

% sum_mono
tff(fact_6139_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),K5))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5))) ) ).

% sum_mono
tff(fact_6140_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cq(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).

% sum.distrib
tff(fact_6141_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_cp(G),H)),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),A2)),aa_set_nat_real(groups6591440286371151544t_real(H),A2)) ).

% sum.distrib
tff(fact_6142_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_gd(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_6143_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ge(G),H),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).

% sum.distrib
tff(fact_6144_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
          & ( B != A ) ) ) ).

% gcd_nat.asym
tff(fact_6145_gcd__nat_Orefl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% gcd_nat.refl
tff(fact_6146_gcd__nat_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% gcd_nat.trans
tff(fact_6147_gcd__nat_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% gcd_nat.eq_iff
tff(fact_6148_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A))
        & ( A != A ) ) ).

% gcd_nat.irrefl
tff(fact_6149_gcd__nat_Oantisym,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( A = B ) ) ) ).

% gcd_nat.antisym
tff(fact_6150_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
          & ( B != C ) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_6151_gcd__nat_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
          & ( B != C ) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_6152_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_6153_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_6154_gcd__nat_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
          & ( A != B ) )
        | ( A = B ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_6155_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_6156_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% gcd_nat.strict_implies_order
tff(fact_6157_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_6158_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_6159_dvd__refl,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).

% dvd_refl
tff(fact_6160_dvd__refl,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).

% dvd_refl
tff(fact_6161_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_trans
tff(fact_6162_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_trans
tff(fact_6163_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N2),M))
       => ( M = N2 ) ) ) ).

% dvd_antisym
tff(fact_6164_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups5808333547571424918x_real(G,A2) != zero_zero_real )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A2))
           => ( aa_complex_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6165_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups8097168146408367636l_real(G,A2) != zero_zero_real )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6166_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups8778361861064173332t_real(G,A2) != zero_zero_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6167_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A2: set_complex] :
      ( ( groups5058264527183730370ex_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A2))
           => ( aa_complex_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6168_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups1300246762558778688al_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6169_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A2: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6170_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A2: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A2) != zero_zero_rat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6171_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_nat,A2: set_complex] :
      ( ( groups5693394587270226106ex_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A2))
           => ( aa_complex_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6172_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A2: set_real] :
      ( ( groups1935376822645274424al_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6173_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A2: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A2) != zero_zero_nat )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_nat(G,A3) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6174_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_nat(G,X4) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_6175_sum_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_real(G,X4) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A2) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_6176_sum_Oneutral,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => ( aa_complex_complex(G,X4) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A2) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_6177_sum_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( aa_int_int(G,X4) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A2) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_6178_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,T: real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N)))
     => ( pp(aa_real_bool(sums_real(F),S))
       => ( pp(aa_real_bool(sums_real(G),T))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,S),T)) ) ) ) ).

% sums_le
tff(fact_6179_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N)))
     => ( pp(aa_nat_bool(sums_nat(F),S))
       => ( pp(aa_nat_bool(sums_nat(G),T))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T)) ) ) ) ).

% sums_le
tff(fact_6180_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,T: int] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N)))
     => ( pp(aa_int_bool(sums_int(F),S))
       => ( pp(aa_int_bool(sums_int(G),T))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S),T)) ) ) ) ).

% sums_le
tff(fact_6181_sums__0,axiom,
    ! [F: fun_nat_real] :
      ( ! [N: nat] : aa_nat_real(F,N) = zero_zero_real
     => pp(aa_real_bool(sums_real(F),zero_zero_real)) ) ).

% sums_0
tff(fact_6182_sums__0,axiom,
    ! [F: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_nat(F,N) = zero_zero_nat
     => pp(aa_nat_bool(sums_nat(F),zero_zero_nat)) ) ).

% sums_0
tff(fact_6183_sums__0,axiom,
    ! [F: fun_nat_int] :
      ( ! [N: nat] : aa_nat_int(F,N) = zero_zero_int
     => pp(aa_int_bool(sums_int(F),zero_zero_int)) ) ).

% sums_0
tff(fact_6184_sums__finite,axiom,
    ! [N5: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => pp(aa_int_bool(sums_int(F),groups3539618377306564664at_int(F,N5))) ) ) ).

% sums_finite
tff(fact_6185_sums__finite,axiom,
    ! [N5: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => pp(aa_nat_bool(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N5))) ) ) ).

% sums_finite
tff(fact_6186_sums__finite,axiom,
    ! [N5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => pp(aa_real_bool(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N5))) ) ) ).

% sums_finite
tff(fact_6187_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ck(P),F)),groups3539618377306564664at_int(F,collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6188_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cj(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6189_sums__If__finite,axiom,
    ! [P: fun_nat_bool,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(P)))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ci(P),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P)))) ) ).

% sums_If_finite
tff(fact_6190_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_cn(A2),F)),groups3539618377306564664at_int(F,A2))) ) ).

% sums_If_finite_set
tff(fact_6191_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cm(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2))) ) ).

% sums_If_finite_set
tff(fact_6192_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_cl(A2),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A2))) ) ).

% sums_If_finite_set
tff(fact_6193_dvd__0__left,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_6194_dvd__0__left,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_6195_dvd__0__left,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_6196_dvd__0__left,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_6197_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_6198_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_6199_dvd__productE,axiom,
    ! [P3: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Y3) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X4),A))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_6200_dvd__productE,axiom,
    ! [P3: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P3),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
     => ~ ! [X4: int,Y3: int] :
            ( ( P3 = aa_int_int(aa_int_fun_int_int(times_times_int,X4),Y3) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X4),A))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y3),B)) ) ) ) ).

% dvd_productE
tff(fact_6201_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
     => ? [B7: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C5) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B7),B))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C5),C)) ) ) ).

% division_decomp
tff(fact_6202_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
     => ? [B7: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B7),C5) )
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B7),B))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C5),C)) ) ) ).

% division_decomp
tff(fact_6203_dvdE,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvdE
tff(fact_6204_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ~ ! [K2: rat] : A != aa_rat_rat(times_times_rat(B),K2) ) ).

% dvdE
tff(fact_6205_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_6206_dvdE,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvdE
tff(fact_6207_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).

% dvdI
tff(fact_6208_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).

% dvdI
tff(fact_6209_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% dvdI
tff(fact_6210_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% dvdI
tff(fact_6211_dvd__def,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_6212_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_6213_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6214_dvd__def,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6215_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult
tff(fact_6216_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult
tff(fact_6217_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).

% dvd_mult
tff(fact_6218_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ).

% dvd_mult
tff(fact_6219_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).

% dvd_mult2
tff(fact_6220_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).

% dvd_mult2
tff(fact_6221_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).

% dvd_mult2
tff(fact_6222_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ).

% dvd_mult2
tff(fact_6223_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ).

% dvd_mult_left
tff(fact_6224_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ).

% dvd_mult_left
tff(fact_6225_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ).

% dvd_mult_left
tff(fact_6226_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ).

% dvd_mult_left
tff(fact_6227_dvd__triv__left,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ).

% dvd_triv_left
tff(fact_6228_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B))) ).

% dvd_triv_left
tff(fact_6229_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ).

% dvd_triv_left
tff(fact_6230_dvd__triv__left,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ).

% dvd_triv_left
tff(fact_6231_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),D))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_6232_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),D))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ).

% mult_dvd_mono
tff(fact_6233_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_6234_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ).

% mult_dvd_mono
tff(fact_6235_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C)) ) ).

% dvd_mult_right
tff(fact_6236_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C)) ) ).

% dvd_mult_right
tff(fact_6237_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ).

% dvd_mult_right
tff(fact_6238_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ).

% dvd_mult_right
tff(fact_6239_dvd__triv__right,axiom,
    ! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))) ).

% dvd_triv_right
tff(fact_6240_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A))) ).

% dvd_triv_right
tff(fact_6241_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))) ).

% dvd_triv_right
tff(fact_6242_dvd__triv__right,axiom,
    ! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))) ).

% dvd_triv_right
tff(fact_6243_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6244_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6245_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6246_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_add_right_iff
tff(fact_6247_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6248_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6249_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6250_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).

% dvd_add_left_iff
tff(fact_6251_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ) ).

% dvd_add
tff(fact_6252_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))) ) ) ).

% dvd_add
tff(fact_6253_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ) ).

% dvd_add
tff(fact_6254_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ) ).

% dvd_add
tff(fact_6255_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).

% dvd_unit_imp_unit
tff(fact_6256_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int)) ) ) ).

% dvd_unit_imp_unit
tff(fact_6257_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% unit_imp_dvd
tff(fact_6258_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% unit_imp_dvd
tff(fact_6259_one__dvd,axiom,
    ! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,one_one_complex),A)) ).

% one_dvd
tff(fact_6260_one__dvd,axiom,
    ! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).

% one_dvd
tff(fact_6261_one__dvd,axiom,
    ! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,one_one_rat),A)) ).

% one_dvd
tff(fact_6262_one__dvd,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A)) ).

% one_dvd
tff(fact_6263_one__dvd,axiom,
    ! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A)) ).

% one_dvd
tff(fact_6264_dvd__diff,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Z2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z2))) ) ) ).

% dvd_diff
tff(fact_6265_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Z2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_rat_rat(minus_minus_rat(Y),Z2))) ) ) ).

% dvd_diff
tff(fact_6266_dvd__diff,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z2))) ) ) ).

% dvd_diff
tff(fact_6267_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6268_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_6269_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6270_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(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_6271_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) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6272_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) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6273_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) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6274_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
       => ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),B))
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6275_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_6276_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_6277_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_6278_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat))
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_6279_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_6280_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_6281_gcd__nat_Oextremum,axiom,
    ! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).

% gcd_nat.extremum
tff(fact_6282_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6283_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6284_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6285_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),K)) ) ).

% dvd_if_abs_eq
tff(fact_6286_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_6287_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_6288_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).

% dvd_mod_iff
tff(fact_6289_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6290_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(modulo_modulo_int(A),B)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6291_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6292_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6293_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6294_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),zero_zero_real)) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real)) ) ).

% sum_nonpos
tff(fact_6295_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6296_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6297_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6298_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),zero_zero_rat)) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat)) ) ).

% sum_nonpos
tff(fact_6299_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_6300_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_6301_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),zero_zero_nat)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat)) ) ).

% sum_nonpos
tff(fact_6302_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6303_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6304_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2))) ) ).

% sum_nonneg
tff(fact_6305_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6306_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6307_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6308_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2))) ) ).

% sum_nonneg
tff(fact_6309_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_6310_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_6311_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2))) ) ).

% sum_nonneg
tff(fact_6312_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))) ) ) ).

% dvd_diff_nat
tff(fact_6313_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6314_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
       => ( pp(aa_set_nat_bool(member_nat(I),I5))
         => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6315_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6316_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6317_sum__mono__inv,axiom,
    ! [F: fun_Ex4152069839811072081at_rat,I5: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,I: extended_enat] :
      ( ( groups1392844769737527556at_rat(F,I5) = groups1392844769737527556at_rat(G,I5) )
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I2)),aa_Extended_enat_rat(G,I2))) )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
           => ( aa_Extended_enat_rat(F,I) = aa_Extended_enat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6318_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6319_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
       => ( pp(aa_set_int_bool(member_int(I),I5))
         => ( pp(aa_set_int_bool(finite_finite_int,I5))
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6320_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) )
       => ( pp(aa_set_complex_bool(member_complex(I),I5))
         => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6321_sum__mono__inv,axiom,
    ! [F: fun_Ex7660298398204468809at_nat,I5: set_Extended_enat,G: fun_Ex7660298398204468809at_nat,I: extended_enat] :
      ( ( groups2027974829824023292at_nat(F,I5) = groups2027974829824023292at_nat(G,I5) )
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,I2)),aa_Extended_enat_nat(G,I2))) )
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
         => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
           => ( aa_Extended_enat_nat(F,I) = aa_Extended_enat_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6322_sum__mono__inv,axiom,
    ! [F: fun_real_int,I5: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I5) = groups1932886352136224148al_int(G,I5) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2))) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6323_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( 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),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_6324_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X4)),A2))
           => ( aa_nat_real(F,aa_nat_nat(suc,X4)) = aa_nat_real(G,aa_nat_nat(suc,X4)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A2) = aa_set_nat_real(groups6591440286371151544t_real(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_6325_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_cb(I),F)),aa_nat_real(F,I))) ).

% sums_single
tff(fact_6326_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cc(I),F)),aa_nat_nat(F,I))) ).

% sums_single
tff(fact_6327_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_cd(I),F)),aa_nat_int(F,I))) ).

% sums_single
tff(fact_6328_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( pp(aa_real_bool(sums_real(F),A))
     => ( pp(aa_real_bool(sums_real(G),B))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_cp(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).

% sums_add
tff(fact_6329_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( pp(aa_nat_bool(sums_nat(F),A))
     => ( pp(aa_nat_bool(sums_nat(G),B))
       => pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cq(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% sums_add
tff(fact_6330_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( pp(aa_int_bool(sums_int(F),A))
     => ( pp(aa_int_bool(sums_int(G),B))
       => pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_cr(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% sums_add
tff(fact_6331_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_gf(A2),P))) = groups8097168146408367636l_real(aa_fun6348708079608492509l_real(aTP_Lamm_gg(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6332_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_gh(A2),P))) = groups8778361861064173332t_real(aa_fun6974987029372589917t_real(aTP_Lamm_gi(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6333_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_gj(A2),P))) = groups5808333547571424918x_real(aa_fun2902006202505214429x_real(aTP_Lamm_gk(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6334_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4148127829035722712t_real(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_gl(A2),P))) = groups4148127829035722712t_real(aa_fun1090353380788292083t_real(aTP_Lamm_gm(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6335_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_gf(A2),P))) = groups1300246762558778688al_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_gn(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6336_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_go(A2),P))) = groups2906978787729119204at_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_gp(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6337_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_gh(A2),P))) = groups3906332499630173760nt_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_gq(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6338_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_rat,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_gj(A2),P))) = groups5058264527183730370ex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_gr(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6339_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups1392844769737527556at_rat(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_gl(A2),P))) = groups1392844769737527556at_rat(aa_fun6658670135733114143at_rat(aTP_Lamm_gs(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6340_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_nat,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups1935376822645274424al_nat(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_gf(A2),P))) = groups1935376822645274424al_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_gt(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6341_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S3: real,A2: set_nat,S4: real,F: fun_nat_real] :
      ( pp(aa_real_bool(sums_real(G),S3))
     => ( pp(aa_set_nat_bool(finite_finite_nat,A2))
       => ( ( S4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gu(G),F)),A2)) )
         => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_gv(G),A2),F)),S4)) ) ) ) ).

% sums_If_finite_set'
tff(fact_6342_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gw,A)),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gw,B)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).

% subset_divisors_dvd
tff(fact_6343_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_gx,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_gx,B))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).

% subset_divisors_dvd
tff(fact_6344_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gy,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gy,B))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% subset_divisors_dvd
tff(fact_6345_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gw,A)),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_gw,B)))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
        & ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6346_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gy,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gy,B))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6347_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_gx,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_gx,B))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6348_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( finite_finite_real(A2)
     => ( ! [X4: real] :
            ( member_real(X4,A2)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6349_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6350_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A2))
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6351_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
       => ( ( groups4148127829035722712t_real(F,A2) = zero_zero_real )
        <=> ! [X3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
             => ( aa_Ext2541572767299617110t_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6352_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( ! [X4: real] :
            ( member_real(X4,A2)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6353_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4))) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( pp(aa_set_nat_bool(member_nat(X3),A2))
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6354_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: int] :
              ( pp(aa_set_int_bool(member_int(X3),A2))
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6355_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: complex] :
              ( pp(aa_set_complex_bool(member_complex(X3),A2))
             => ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6356_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X4))) )
       => ( ( groups1392844769737527556at_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
             => ( aa_Extended_enat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6357_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( ! [X4: real] :
            ( member_real(X4,A2)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
       => ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6358_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_int(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6359_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_int(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_complex_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6360_sum__le__included,axiom,
    ! [S: set_int,T: set_Extended_enat,G: fun_Ex1451620480498224165t_real,I: fun_Ex3482447378695272101at_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G,X4))) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Extended_enat_int(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_Ext2541572767299617110t_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups4148127829035722712t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6361_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_complex(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_int_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6362_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_complex_complex(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6363_sum__le__included,axiom,
    ! [S: set_complex,T: set_Extended_enat,G: fun_Ex1451620480498224165t_real,I: fun_Ex3741802288643809319omplex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G,X4))) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Ext8748583412541189208omplex(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_Ext2541572767299617110t_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups4148127829035722712t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6364_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_int,G: fun_int_real,I: fun_in5914218760682845459d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set_int_bool(finite_finite_int,T))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X4))) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
               => ? [Xa: int] :
                    ( pp(aa_set_int_bool(member_int(Xa),T))
                    & ( aa_int_Extended_enat(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_int_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6365_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_complex,G: fun_complex_real,I: fun_co6496365337113880337d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X4))) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
               => ? [Xa: complex] :
                    ( pp(aa_set_complex_bool(member_complex(Xa),T))
                    & ( aa_com3683005661447327866d_enat(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_complex_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6366_sum__le__included,axiom,
    ! [S: set_Extended_enat,T: set_Extended_enat,G: fun_Ex1451620480498224165t_real,I: fun_Ex5852502975177633731d_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),T))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(G,X4))) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S))
               => ? [Xa: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa),T))
                    & ( aa_Ext7969882492699152700d_enat(I,Xa) = X4 )
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,Xa))) ) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,S)),groups4148127829035722712t_real(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6367_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( pp(aa_set_nat_bool(finite_finite_nat,T))
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),T))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X4))) )
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S))
               => ? [Xa: nat] :
                    ( pp(aa_set_nat_bool(member_nat(Xa),T))
                    & ( aa_nat_nat(I,Xa) = X4 )
                    & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,Xa))) ) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).

% sum_le_included
tff(fact_6368_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X2)),aa_int_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6369_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6370_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
       => ( ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X2)),aa_Ext2541572767299617110t_real(G,X2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6371_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [X4: nat] :
            ( pp(aa_set_nat_bool(member_nat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
       => ( ? [X2: nat] :
              ( pp(aa_set_nat_bool(member_nat(X2),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6372_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6373_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6374_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,X4)),aa_Extended_enat_rat(G,X4))) )
       => ( ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X2)),aa_Extended_enat_rat(G,X2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A2)),groups1392844769737527556at_rat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6375_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [X4: int] :
            ( pp(aa_set_int_bool(member_int(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
       => ( ? [X2: int] :
              ( pp(aa_set_int_bool(member_int(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6376_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [X4: complex] :
            ( pp(aa_set_complex_bool(member_complex(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
       => ( ? [X2: complex] :
              ( pp(aa_set_complex_bool(member_complex(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6377_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [X4: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Extended_enat_nat(F,X4)),aa_Extended_enat_nat(G,X4))) )
       => ( ? [X2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Extended_enat_nat(F,X2)),aa_Extended_enat_nat(G,X2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups2027974829824023292at_nat(F,A2)),groups2027974829824023292at_nat(G,A2))) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6378_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6379_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6380_sum_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S3: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,zero_zero_real),zero_zero_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_Ext2541572767299617110t_real(H,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups4148127829035722712t_real(H,S3)),groups4148127829035722712t_real(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6381_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6382_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_int_rat(H,X4)),aa_int_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups3906332499630173760nt_rat(H,S3)),groups3906332499630173760nt_rat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6383_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_complex_rat(H,X4)),aa_complex_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6384_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S3: set_Extended_enat,H: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,zero_zero_rat),zero_zero_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_Extended_enat_rat(H,X4)),aa_Extended_enat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups1392844769737527556at_rat(H,S3)),groups1392844769737527556at_rat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6385_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_int_nat(H,X4)),aa_int_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6386_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_complex_nat(H,X4)),aa_complex_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6387_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_bool,S3: set_Extended_enat,H: fun_Ex7660298398204468809at_nat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,zero_zero_nat),zero_zero_nat))
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X1),X23))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R,Y1),Y23)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,aa_Extended_enat_nat(H,X4)),aa_Extended_enat_nat(G,X4))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,groups2027974829824023292at_nat(H,S3)),groups2027974829824023292at_nat(G,S3))) ) ) ) ) ).

% sum.related
tff(fact_6388_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6389_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6390_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( member_real(X4,A2)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),aa_real_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6391_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6392_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6393_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,X4)),aa_Extended_enat_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A2)),groups1392844769737527556at_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6394_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( member_real(X4,A2)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6395_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( pp(aa_set_nat_bool(member_nat(X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6396_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6397_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).

% sum_strict_mono
tff(fact_6398_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [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(T4,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T4,T3))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T4,T3))
                   => 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,T3)
                       => ( 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,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6399_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T4: set_int,G: fun_real_real,H: fun_int_real] :
      ( finite_finite_real(S4)
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [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))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => 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] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( 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,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6400_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S3: set_real,I: fun_complex_real,J: fun_real_complex,T4: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( finite_finite_real(S4)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [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))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => 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] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( 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,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6401_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_Extended_enat,S3: set_real,I: fun_Ex1451620480498224165t_real,J: fun_re6065469302365115155d_enat,T4: set_Extended_enat,G: fun_real_real,H: fun_Ex1451620480498224165t_real] :
      ( finite_finite_real(S4)
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_Ext2541572767299617110t_real(I,aa_rea1194391308913012860d_enat(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_rea1194391308913012860d_enat(J,A3)),minus_925952699566721837d_enat(T4,T3))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                 => ( aa_rea1194391308913012860d_enat(J,aa_Ext2541572767299617110t_real(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                   => member_real(aa_Ext2541572767299617110t_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T3))
                       => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_Ext2541572767299617110t_real(H,aa_rea1194391308913012860d_enat(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups4148127829035722712t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6402_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T4: set_real,G: fun_int_real,H: fun_real_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( finite_finite_real(T3)
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T4,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T4,T3))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T4,T3))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(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,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6403_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T4: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(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,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6404_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_complex,S3: set_int,I: fun_complex_int,J: fun_int_complex,T4: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(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,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6405_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_Extended_enat,S3: set_int,I: fun_Ex3482447378695272101at_int,J: fun_in5914218760682845459d_enat,T4: set_Extended_enat,G: fun_int_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_Extended_enat_int(I,aa_int_Extended_enat(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_int_Extended_enat(J,A3)),minus_925952699566721837d_enat(T4,T3))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                 => ( aa_int_Extended_enat(J,aa_Extended_enat_int(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_Extended_enat_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T3))
                       => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_Ext2541572767299617110t_real(H,aa_int_Extended_enat(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups4148127829035722712t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6406_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_real,S3: set_complex,I: fun_real_complex,J: fun_complex_real,T4: set_real,G: fun_complex_real,H: fun_real_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
     => ( finite_finite_real(T3)
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
             => ( aa_real_complex(I,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
               => member_real(aa_complex_real(J,A3),minus_minus_set_real(T4,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T4,T3))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T4,T3))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I,B3)),minus_811609699411566653omplex(S3,S4))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S4))
                     => ( aa_complex_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S3))
                         => ( aa_real_real(H,aa_complex_real(J,A3)) = aa_complex_real(G,A3) ) )
                     => ( groups5808333547571424918x_real(G,S3) = groups8097168146408367636l_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6407_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_int,S3: set_complex,I: fun_int_complex,J: fun_complex_int,T4: set_int,G: fun_complex_real,H: fun_int_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
             => ( aa_int_complex(I,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J,A3)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_int_complex(I,B3)),minus_811609699411566653omplex(S3,S4))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S4))
                     => ( aa_complex_real(G,A3) = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_real(H,B3) = zero_zero_real ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S3))
                         => ( aa_int_real(H,aa_complex_int(J,A3)) = aa_complex_real(G,A3) ) )
                     => ( groups5808333547571424918x_real(G,S3) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6408_not__is__unit__0,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),one_one_nat)) ).

% not_is_unit_0
tff(fact_6409_not__is__unit__0,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),one_one_int)) ).

% not_is_unit_0
tff(fact_6410_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% pinf(9)
tff(fact_6411_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))) ) ) ).

% pinf(9)
tff(fact_6412_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% pinf(9)
tff(fact_6413_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% pinf(9)
tff(fact_6414_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X2))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% pinf(10)
tff(fact_6415_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X2))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))) ) ) ).

% pinf(10)
tff(fact_6416_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X2))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% pinf(10)
tff(fact_6417_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X2))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% pinf(10)
tff(fact_6418_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% minf(9)
tff(fact_6419_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))) ) ) ).

% minf(9)
tff(fact_6420_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% minf(9)
tff(fact_6421_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% minf(9)
tff(fact_6422_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X2),Z))
     => ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S)))
      <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),S))) ) ) ).

% minf(10)
tff(fact_6423_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X2),Z))
     => ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)))
      <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))) ) ) ).

% minf(10)
tff(fact_6424_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)))
      <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))) ) ) ).

% minf(10)
tff(fact_6425_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)))
      <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))) ) ) ).

% minf(10)
tff(fact_6426_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6427_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6428_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6429_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6430_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6431_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat)) ) ) ).

% is_unit_mult_iff
tff(fact_6432_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int)) ) ) ).

% is_unit_mult_iff
tff(fact_6433_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_6434_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff
tff(fact_6435_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_6436_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% mult_unit_dvd_iff
tff(fact_6437_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6438_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6439_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6440_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6441_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6442_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6443_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6444_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6445_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6446_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6447_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6448_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6449_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,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_6450_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,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_6451_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),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_6452_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),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_6453_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))) ) ).

% dvd_mult_imp_div
tff(fact_6454_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C))) ) ).

% dvd_mult_imp_div
tff(fact_6455_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,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_6456_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,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_6457_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6458_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6459_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6460_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6461_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_6462_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(C),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% dvd_div_unit_iff
tff(fact_6463_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_6464_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).

% div_unit_dvd_iff
tff(fact_6465_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6466_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6467_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(divide_divide_int(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_6468_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,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_6469_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(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_6470_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_6471_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_6472_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => ( aa_int_int(divide_divide_int(A),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_6473_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),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_6474_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(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_6475_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_6476_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_6477_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_gz(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult2_iff
tff(fact_6478_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_cf(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)))
      <=> pp(aa_real_bool(sums_real(F),D)) ) ) ).

% sums_mult_iff
tff(fact_6479_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_6480_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_6481_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_0_imp_dvd
tff(fact_6482_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6483_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
    <=> ( aa_int_int(modulo_modulo_int(B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6484_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6485_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_6486_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_6487_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).

% mod_eq_0_iff_dvd
tff(fact_6488_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M))) ) ) ).

% dvd_power_le
tff(fact_6489_dvd__power__le,axiom,
    ! [X: real,Y: real,N2: nat,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(X),N2)),aa_nat_real(power_power_real(Y),M))) ) ) ).

% dvd_power_le
tff(fact_6490_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N2)),aa_nat_int(power_power_int(Y),M))) ) ) ).

% dvd_power_le
tff(fact_6491_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N2: nat,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N2)),aa_nat_complex(power_power_complex(Y),M))) ) ) ).

% dvd_power_le
tff(fact_6492_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_6493_power__le__dvd,axiom,
    ! [A: real,N2: nat,B: real,M: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),N2)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_6494_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N2)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_6495_power__le__dvd,axiom,
    ! [A: complex,N2: nat,B: complex,M: nat] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),N2)),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),B)) ) ) ).

% power_le_dvd
tff(fact_6496_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2))) ) ).

% le_imp_power_dvd
tff(fact_6497_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(power_power_real(A),M)),aa_nat_real(power_power_real(A),N2))) ) ).

% le_imp_power_dvd
tff(fact_6498_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N2))) ) ).

% le_imp_power_dvd
tff(fact_6499_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N2))) ) ).

% le_imp_power_dvd
tff(fact_6500_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6501_dvd__minus__mod,axiom,
    ! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(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_6502_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)))) ).

% dvd_minus_mod
tff(fact_6503_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N2),M)) ) ) ).

% nat_dvd_not_less
tff(fact_6504_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).

% dvd_pos_nat
tff(fact_6505_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2)) ) ) ).

% dvd_minus_self
tff(fact_6506_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_real,B2: real,I: real] :
      ( finite_finite_real(S)
     => ( ! [I2: real] :
            ( member_real(I2,S)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S) = B2 )
         => ( member_real(I,S)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6507_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_real,B2: real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6508_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_real,B2: real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6509_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real,B2: real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
       => ( ( groups4148127829035722712t_real(F,S) = B2 )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6510_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_rat,B2: rat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I2: real] :
            ( member_real(I2,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S) = B2 )
         => ( member_real(I,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6511_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S) = B2 )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6512_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: fun_int_rat,B2: rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = B2 )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6513_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = B2 )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6514_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,B2: rat,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I2))) )
       => ( ( groups1392844769737527556at_rat(F,S) = B2 )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_Extended_enat_rat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6515_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: fun_real_nat,B2: nat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I2: real] :
            ( member_real(I2,S)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
       => ( ( groups1935376822645274424al_nat(F,S) = B2 )
         => ( member_real(I,S)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I)),B2)) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6516_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_real,I: real] :
      ( finite_finite_real(S)
     => ( ! [I2: real] :
            ( member_real(I2,S)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
         => ( member_real(I,S)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6517_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_real,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6518_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_real,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6519_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: fun_Ex1451620480498224165t_real,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),S))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
       => ( ( groups4148127829035722712t_real(F,S) = zero_zero_real )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => ( aa_Ext2541572767299617110t_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6520_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_rat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I2: real] :
            ( member_real(I2,S)
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
       => ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
         => ( member_real(I,S)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6521_sum__nonneg__0,axiom,
    ! [S: set_nat,F: fun_nat_rat,I: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
       => ( ( groups2906978787729119204at_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_nat_bool(member_nat(I),S))
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6522_sum__nonneg__0,axiom,
    ! [S: set_int,F: fun_int_rat,I: int] :
      ( pp(aa_set_int_bool(finite_finite_int,S))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
       => ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_int_bool(member_int(I),S))
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6523_sum__nonneg__0,axiom,
    ! [S: set_complex,F: fun_complex_rat,I: complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
       => ( ( groups5058264527183730370ex_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set_complex_bool(member_complex(I),S))
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6524_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,I: extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),S))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I2))) )
       => ( ( groups1392844769737527556at_rat(F,S) = zero_zero_rat )
         => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),S))
           => ( aa_Extended_enat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6525_sum__nonneg__0,axiom,
    ! [S: set_real,F: fun_real_nat,I: real] :
      ( finite_finite_real(S)
     => ( ! [I2: real] :
            ( member_real(I2,S)
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
       => ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
         => ( member_real(I,S)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6526_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).

% dvd_diffD
tff(fact_6527_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N2)) ) ) ) ).

% dvd_diffD1
tff(fact_6528_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_6529_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X4: nat,Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6530_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),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)),D5) )
        | ( 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)),D5) ) ) ) ).

% bezout_add_nat
tff(fact_6531_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),A))
      & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),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)) = D5 )
        | ( 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)) = D5 ) ) ) ).

% bezout1_nat
tff(fact_6532_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ha(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6533_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_hb(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6534_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_hc(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6535_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_hd(G)))) = groups4148127829035722712t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6536_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_he(G)))) = groups1300246762558778688al_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6537_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_hf(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6538_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_hg(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6539_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups1392844769737527556at_rat(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_hh(G)))) = groups1392844769737527556at_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6540_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_hi(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6541_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_hj(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6542_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6543_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6544_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hl(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6545_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ce(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6546_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6547_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6548_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6549_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I)))
         => ( ! [I2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups4148127829035722712t_real(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6550_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I)))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6551_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I)))
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6552_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6553_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6554_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,I)))
         => ( ! [I2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1392844769737527556at_rat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6555_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_nat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I)))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5))) ) ) ) ) ).

% sum_pos2
tff(fact_6556_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6557_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups4148127829035722712t_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6558_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6559_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6560_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6561_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1392844769737527556at_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6562_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6563_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6564_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6565_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5))) ) ) ) ).

% sum_pos
tff(fact_6566_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6567_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,T4) = groups8778361861064173332t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6568_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,T4) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6569_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,T4) = groups4148127829035722712t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6570_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6571_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups3906332499630173760nt_rat(G,T4) = groups3906332499630173760nt_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6572_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,T4) = groups5058264527183730370ex_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6573_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_rat(G,X4) = zero_zero_rat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Extended_enat_rat(G,X4) = aa_Extended_enat_rat(H,X4) ) )
           => ( groups1392844769737527556at_rat(G,T4) = groups1392844769737527556at_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6574_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) = groups1935376822645274424al_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6575_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,T4) = groups4541462559716669496nt_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6576_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6577_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6578_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6579_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(H,X4) = zero_zero_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups4148127829035722712t_real(G,S3) = groups4148127829035722712t_real(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6580_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6581_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6582_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_rat(G,X4) = aa_complex_rat(H,X4) ) )
           => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6583_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_rat(H,X4) = zero_zero_rat ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Extended_enat_rat(G,X4) = aa_Extended_enat_rat(H,X4) ) )
           => ( groups1392844769737527556at_rat(G,S3) = groups1392844769737527556at_rat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6584_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6585_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(H,X4) = zero_zero_nat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_nat(G,X4) = aa_int_nat(H,X4) ) )
           => ( groups4541462559716669496nt_nat(G,S3) = groups4541462559716669496nt_nat(H,T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6586_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T4) = groups8778361861064173332t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6587_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T4) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6588_sum_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,T4) = groups4148127829035722712t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6589_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T4) = groups3906332499630173760nt_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6590_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T4) = groups5058264527183730370ex_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6591_sum_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_rat(G,X4) = zero_zero_rat ) )
         => ( groups1392844769737527556at_rat(G,T4) = groups1392844769737527556at_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6592_sum_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T4) = groups4541462559716669496nt_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6593_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T4) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6594_sum_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,T4) = groups2027974829824023292at_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6595_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T4) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6596_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G,X4) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6597_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G,X4) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6598_sum_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = zero_zero_real ) )
         => ( groups4148127829035722712t_real(G,S3) = groups4148127829035722712t_real(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6599_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G,X4) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6600_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G,X4) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6601_sum_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_rat(G,X4) = zero_zero_rat ) )
         => ( groups1392844769737527556at_rat(G,S3) = groups1392844769737527556at_rat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6602_sum_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(G,X4) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S3) = groups4541462559716669496nt_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6603_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_nat(G,X4) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6604_sum_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_nat(G,X4) = zero_zero_nat ) )
         => ( groups2027974829824023292at_nat(G,S3) = groups2027974829824023292at_nat(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6605_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_int(G,X4) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T4) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6606_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6607_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) )
               => ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6608_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6609_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext2541572767299617110t_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) )
               => ( groups4148127829035722712t_real(G,A2) = groups4148127829035722712t_real(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6610_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6611_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) )
               => ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6612_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6613_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Extended_enat_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1392844769737527556at_rat(G,C4) = groups1392844769737527556at_rat(H,C4) )
               => ( groups1392844769737527556at_rat(G,A2) = groups1392844769737527556at_rat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6614_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
               => ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6615_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) )
               => ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6616_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6617_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_real(H,B3) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) )
              <=> ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6618_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6619_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext2541572767299617110t_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = zero_zero_real ) )
             => ( ( groups4148127829035722712t_real(G,A2) = groups4148127829035722712t_real(H,B2) )
              <=> ( groups4148127829035722712t_real(G,C4) = groups4148127829035722712t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6620_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6621_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) )
              <=> ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6622_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6623_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat,H: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Extended_enat_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Extended_enat_rat(H,B3) = zero_zero_rat ) )
             => ( ( groups1392844769737527556at_rat(G,A2) = groups1392844769737527556at_rat(H,B2) )
              <=> ( groups1392844769737527556at_rat(G,C4) = groups1392844769737527556at_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6624_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) )
              <=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6625_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_nat(H,B3) = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) )
              <=> ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6626_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6627_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6628_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups4148127829035722712t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups4148127829035722712t_real(G,minus_925952699566721837d_enat(A2,B2))),groups4148127829035722712t_real(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6629_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2))),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6630_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2))),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6631_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups1392844769737527556at_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups1392844769737527556at_rat(G,minus_925952699566721837d_enat(A2,B2))),groups1392844769737527556at_rat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6632_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
     => ( pp(aa_set_int_bool(finite_finite_int,A2))
       => ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6633_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6634_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),A2))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
       => ( groups2027974829824023292at_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups2027974829824023292at_nat(G,minus_925952699566721837d_enat(A2,B2))),groups2027974829824023292at_nat(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6635_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).

% sum.subset_diff
tff(fact_6636_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_cf(C),F)),A))
     => ( ( C != zero_zero_real )
       => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).

% sums_mult_D
tff(fact_6637_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hm(C),F)),A))
     => ( ( C != zero_zero_complex )
       => pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ) ).

% sums_mult_D
tff(fact_6638_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( pp(aa_real_bool(sums_real(aTP_Lamm_cs(F)),S))
       => pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_Suc_imp
tff(fact_6639_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_cs(F)),L))
     => pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc
tff(fact_6640_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( pp(aa_nat_bool(sums_nat(aTP_Lamm_hk(F)),L))
     => pp(aa_nat_bool(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_6641_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( pp(aa_int_bool(sums_int(aTP_Lamm_hn(F)),L))
     => pp(aa_int_bool(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_6642_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( pp(aa_real_bool(sums_real(aTP_Lamm_cs(F)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_nat_real(F,zero_zero_nat)))) ) ).

% sums_Suc_iff
tff(fact_6643_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: fun_nat_real,S: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N2))
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ho(N2),F)),S))
      <=> pp(aa_real_bool(sums_real(F),S)) ) ) ).

% sums_zero_iff_shift
tff(fact_6644_unity__coeff__ex,axiom,
    ! [P: fun_real_bool,L: real] :
      ( ? [X3: real] : pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X3)))
    <=> ? [X3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),zero_zero_real)))
          & pp(aa_real_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_6645_unity__coeff__ex,axiom,
    ! [P: fun_rat_bool,L: rat] :
      ( ? [X3: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X3)))
    <=> ? [X3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X3),zero_zero_rat)))
          & pp(aa_rat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_6646_unity__coeff__ex,axiom,
    ! [P: fun_nat_bool,L: nat] :
      ( ? [X3: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3)))
    <=> ? [X3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_6647_unity__coeff__ex,axiom,
    ! [P: fun_int_bool,L: int] :
      ( ? [X3: int] : pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3)))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int)))
          & pp(aa_int_bool(P,X3)) ) ) ).

% unity_coeff_ex
tff(fact_6648_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_6649_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_6650_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_6651_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
       => ( ( aa_int_int(divide_divide_int(B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_6652_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_6653_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(divide_divide_int(A),B)),C))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ) ).

% div_dvd_iff_mult
tff(fact_6654_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_6655_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(divide_divide_int(B),C)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)) ) ) ) ).

% dvd_div_iff_mult
tff(fact_6656_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),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_6657_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),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_6658_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_6659_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(divide_divide_int(A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_6660_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D4))
     => ! [X2: real,K4: real] :
          ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),T)))
        <=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),T))) ) ) ).

% inf_period(4)
tff(fact_6661_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D4))
     => ! [X2: rat,K4: rat] :
          ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),T)))
        <=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D4))),T))) ) ) ).

% inf_period(4)
tff(fact_6662_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X2: int,K4: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
        <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(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,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),T))) ) ) ).

% inf_period(4)
tff(fact_6663_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D4))
     => ! [X2: real,K4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),T)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),T))) ) ) ).

% inf_period(3)
tff(fact_6664_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D4))
     => ! [X2: rat,K4: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),T)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D4))),T))) ) ) ).

% inf_period(3)
tff(fact_6665_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X2: int,K4: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(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,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),T))) ) ) ).

% inf_period(3)
tff(fact_6666_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6667_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_6668_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6669_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6670_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(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_6671_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6672_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),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_6673_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_6674_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_6675_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(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_6676_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_6677_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( ( aa_int_int(divide_divide_int(A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_6678_is__unit__power__iff,axiom,
    ! [A: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),one_one_nat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6679_is__unit__power__iff,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N2)),one_one_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6680_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_6681_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
     => ( aa_int_int(modulo_modulo_int(A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_6682_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_6683_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) ) ) ).

% dvd_imp_le
tff(fact_6684_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2)) ) ) ).

% dvd_mult_cancel
tff(fact_6685_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_6686_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X4: nat,Y3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),A))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D5),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)),D5) ) ) ) ).

% bezout_add_strong_nat
tff(fact_6687_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N2),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_6688_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),Q4) )
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_6689_real__of__nat__div,axiom,
    ! [D: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),N2))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_6690_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_hp(X),M),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),I5)) ).

% sum_power_add
tff(fact_6691_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_hq(X),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_6692_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_hr(X),M),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),I5)) ).

% sum_power_add
tff(fact_6693_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_hs(X),M)),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),I5)) ).

% sum_power_add
tff(fact_6694_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B2,A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6695_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6696_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6697_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B2))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B3))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6698_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B2,A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6699_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6700_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6701_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B2))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,B3))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1392844769737527556at_rat(F,A2)),groups1392844769737527556at_rat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6702_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_nat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B2,A2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6703_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,B3))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2))) ) ) ) ).

% sum_mono2
tff(fact_6704_sums__if_H,axiom,
    ! [G: fun_nat_real,X: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => pp(aa_real_bool(sums_real(aTP_Lamm_ht(G)),X)) ) ).

% sums_if'
tff(fact_6705_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hu(G),N2),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) ).

% sum.atLeastAtMost_rev
tff(fact_6706_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N2: nat,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_hv(G),N2),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) ).

% sum.atLeastAtMost_rev
tff(fact_6707_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y: real] :
      ( pp(aa_real_bool(sums_real(G),X))
     => ( pp(aa_real_bool(sums_real(F),Y))
       => pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hw(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sums_if
tff(fact_6708_suminf__finite,axiom,
    ! [N5: set_nat,F: fun_nat_int] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N5) ) ) ) ).

% suminf_finite
tff(fact_6709_suminf__finite,axiom,
    ! [N5: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N5) ) ) ) ).

% suminf_finite
tff(fact_6710_suminf__finite,axiom,
    ! [N5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,N5))
     => ( ! [N: nat] :
            ( ~ pp(aa_set_nat_bool(member_nat(N),N5))
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N5) ) ) ) ).

% suminf_finite
tff(fact_6711_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gy,M)))) ) ).

% finite_divisors_nat
tff(fact_6712_powser__sums__if,axiom,
    ! [M: nat,Z2: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_hx(M),Z2)),aa_nat_complex(power_power_complex(Z2),M))) ).

% powser_sums_if
tff(fact_6713_powser__sums__if,axiom,
    ! [M: nat,Z2: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_hy(M),Z2)),aa_nat_real(power_power_real(Z2),M))) ).

% powser_sums_if
tff(fact_6714_powser__sums__if,axiom,
    ! [M: nat,Z2: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_hz(M),Z2)),aa_nat_int(power_power_int(Z2),M))) ).

% powser_sums_if
tff(fact_6715_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_bs(A)),aa_nat_complex(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_6716_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_bt(A)),aa_nat_real(A,zero_zero_nat))) ).

% powser_sums_zero
tff(fact_6717_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6718_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6719_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6720_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6721_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6722_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6723_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = 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),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6724_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6725_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6726_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6727_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6728_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6729_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N2))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6730_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N2))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6731_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6732_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6733_even__zero,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat)) ).

% even_zero
tff(fact_6734_even__zero,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int)) ).

% even_zero
tff(fact_6735_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_6736_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_6737_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_6738_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
       => ( aa_int_int(divide_divide_int(A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(divide_divide_int(one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_6739_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B3),one_one_nat))
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6740_is__unitE,axiom,
    ! [A: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),one_one_int))
               => ( ( aa_int_int(divide_divide_int(one_one_int),A) = B3 )
                 => ( ( aa_int_int(divide_divide_int(one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(divide_divide_int(C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6741_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).

% odd_even_add
tff(fact_6742_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).

% odd_even_add
tff(fact_6743_odd__one,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat)) ).

% odd_one
tff(fact_6744_odd__one,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int)) ).

% odd_one
tff(fact_6745_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2)))
      <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),one_one_nat))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ) ).

% dvd_power_iff
tff(fact_6746_dvd__power__iff,axiom,
    ! [X: int,M: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),N2)))
      <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ) ).

% dvd_power_iff
tff(fact_6747_dvd__power,axiom,
    ! [N2: nat,X: rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        | ( X = one_one_rat ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N2))) ) ).

% dvd_power
tff(fact_6748_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        | ( X = one_one_nat ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N2))) ) ).

% dvd_power
tff(fact_6749_dvd__power,axiom,
    ! [N2: nat,X: real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        | ( X = one_one_real ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(power_power_real(X),N2))) ) ).

% dvd_power
tff(fact_6750_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        | ( X = one_one_int ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N2))) ) ).

% dvd_power
tff(fact_6751_dvd__power,axiom,
    ! [N2: nat,X: complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        | ( X = one_one_complex ) )
     => pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),N2))) ) ).

% dvd_power
tff(fact_6752_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( member_real(B,minus_minus_set_real(B2,A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B)))
           => ( ! [X4: real] :
                  ( member_real(X4,B2)
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6753_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,B)))
           => ( ! [X4: int] :
                  ( pp(aa_set_int_bool(member_int(X4),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6754_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6755_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,B: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B2))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(B2,A2)))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,B)))
           => ( ! [X4: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B2))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X4))) )
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups4148127829035722712t_real(F,A2)),groups4148127829035722712t_real(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6756_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( member_real(B,minus_minus_set_real(B2,A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B)))
           => ( ! [X4: real] :
                  ( member_real(X4,B2)
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6757_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,B)))
           => ( ! [X4: int] :
                  ( pp(aa_set_int_bool(member_int(X4),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6758_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B)))
           => ( ! [X4: complex] :
                  ( pp(aa_set_complex_bool(member_complex(X4),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6759_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,B: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B2))
       => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B),minus_925952699566721837d_enat(B2,A2)))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_Extended_enat_rat(F,B)))
           => ( ! [X4: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),B2))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X4))) )
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1392844769737527556at_rat(F,A2)),groups1392844769737527556at_rat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6760_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: fun_real_nat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( member_real(B,minus_minus_set_real(B2,A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,B)))
           => ( ! [X4: real] :
                  ( member_real(X4,B2)
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6761_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,B)))
           => ( ! [X4: int] :
                  ( pp(aa_set_int_bool(member_int(X4),B2))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X4))) )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2))) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6762_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_6763_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% even_even_mod_4_iff
tff(fact_6764_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),M))
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_6765_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),M)),M))
      <=> ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_6766_dvd__minus__add,axiom,
    ! [Q4: nat,N2: nat,R2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),Q4)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q4)))) ) ) ) ).

% dvd_minus_add
tff(fact_6767_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% power_dvd_imp_le
tff(fact_6768_mod__nat__eqI,axiom,
    ! [R2: nat,N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_6769_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6770_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6771_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6772_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6773_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( groups2906978787729119204at_rat(aTP_Lamm_ib(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6774_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( groups3539618377306564664at_int(aTP_Lamm_ic(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6775_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_id(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,N2))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_6776_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X,I2))) )
     => ( ( groups6621422865394947399nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I2),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_ie(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6777_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I2))) )
     => ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I2),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_if(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6778_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I2))) )
     => ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_ig(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6779_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I5))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I2))) )
     => ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2),B))),Delta)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_ih(X),A),I5),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6780_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(X,I2))) )
     => ( ( groups5808333547571424918x_real(X,I5) = one_one_real )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(A,I2)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ii(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6781_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I2))) )
     => ( ( groups8097168146408367636l_real(X,I5) = one_one_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A,I2)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ij(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6782_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),I5))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I2))) )
     => ( ( groups8778361861064173332t_real(X,I5) = one_one_real )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A,I2)),B))),Delta)) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ik(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6783_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(X,I2))) )
     => ( ( groups5058264527183730370ex_rat(X,I5) = one_one_rat )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_il(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6784_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(X,I2))) )
     => ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_im(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6785_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),I5))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I2))) )
     => ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I2)),B))),Delta)) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_in(X),A),I5)),B))),Delta)) ) ) ) ).

% convex_sum_bound_le
tff(fact_6786_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6787_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6788_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),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,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6789_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_6790_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6791_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6792_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6793_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6794_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
    <=> ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6795_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6796_power__mono__odd,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(B),N2))) ) ) ).

% power_mono_odd
tff(fact_6797_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))) ) ) ).

% power_mono_odd
tff(fact_6798_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(B),N2))) ) ) ).

% power_mono_odd
tff(fact_6799_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I5: set_nat] :
      ( summable_int(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N: nat] :
              ( pp(aa_set_nat_bool(member_nat(N),uminus5710092332889474511et_nat(I5)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,I5)),suminf_int(F))) ) ) ) ).

% sum_le_suminf
tff(fact_6800_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I5: set_nat] :
      ( summable_nat(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N: nat] :
              ( pp(aa_set_nat_bool(member_nat(N),uminus5710092332889474511et_nat(I5)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)),suminf_nat(F))) ) ) ) ).

% sum_le_suminf
tff(fact_6801_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I5: set_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(finite_finite_nat,I5))
       => ( ! [N: nat] :
              ( pp(aa_set_nat_bool(member_nat(N),uminus5710092332889474511et_nat(I5)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),I5)),suminf_real(F))) ) ) ) ).

% sum_le_suminf
tff(fact_6802_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% odd_pos
tff(fact_6803_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% dvd_power_iff_le
tff(fact_6804_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6805_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6806_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_6807_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(M),A)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6808_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6809_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(M),A)))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6810_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)))
    <=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6811_oddE,axiom,
    ! [A: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_6812_oddE,axiom,
    ! [A: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_6813_parity__cases,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_6814_parity__cases,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
         => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_6815_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_6816_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ) ).

% mod2_eq_if
tff(fact_6817_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
       => ( aa_int_int(modulo_modulo_int(A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ) ).

% mod2_eq_if
tff(fact_6818_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ) ).

% mod2_eq_if
tff(fact_6819_zero__le__even__power,axiom,
    ! [N2: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N2))) ) ).

% zero_le_even_power
tff(fact_6820_zero__le__even__power,axiom,
    ! [N2: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2))) ) ).

% zero_le_even_power
tff(fact_6821_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N2))) ) ).

% zero_le_even_power
tff(fact_6822_zero__le__odd__power,axiom,
    ! [N2: nat,A: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N2)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% zero_le_odd_power
tff(fact_6823_zero__le__odd__power,axiom,
    ! [N2: nat,A: rat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2)))
      <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% zero_le_odd_power
tff(fact_6824_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_odd_power
tff(fact_6825_zero__le__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(power_power_real(A),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_6826_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_6827_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).

% zero_le_power_eq
tff(fact_6828_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% minus_one_power_iff
tff(fact_6829_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% minus_one_power_iff
tff(fact_6830_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% minus_one_power_iff
tff(fact_6831_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% minus_one_power_iff
tff(fact_6832_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2) = one_one_Code_integer ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% minus_one_power_iff
tff(fact_6833_power__mono__even,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N2))) ) ) ).

% power_mono_even
tff(fact_6834_power__mono__even,axiom,
    ! [N2: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),aa_nat_real(power_power_real(B),N2))) ) ) ).

% power_mono_even
tff(fact_6835_power__mono__even,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(B),N2))) ) ) ).

% power_mono_even
tff(fact_6836_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(B),N2))) ) ) ).

% power_mono_even
tff(fact_6837_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( groups2906978787729119204at_rat(aTP_Lamm_io(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( groups2906978787729119204at_rat(aTP_Lamm_io(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
tff(fact_6838_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( groups3539618377306564664at_int(aTP_Lamm_ip(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( groups3539618377306564664at_int(aTP_Lamm_ip(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = zero_zero_int ) ) ) ).

% sum_natinterval_diff
tff(fact_6839_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_iq(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_iq(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = zero_zero_real ) ) ) ).

% sum_natinterval_diff
tff(fact_6840_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups2906978787729119204at_rat(aTP_Lamm_ir(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,N2)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_6841_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups3539618377306564664at_int(aTP_Lamm_is(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N2)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_6842_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_it(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N2)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_6843_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E2: real] :
      ( summable_complex(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N6)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_6844_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E2: real] :
      ( summable_real(F)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
                 => ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N6)))),E2)) ) ) ) ).

% summable_partial_sum_bound
tff(fact_6845_geometric__sums,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
     => pp(aa_real_bool(sums_real(power_power_real(C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)))) ) ).

% geometric_sums
tff(fact_6846_geometric__sums,axiom,
    ! [C: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
     => pp(aa_complex_bool(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_6847_power__half__series,axiom,
    pp(aa_real_bool(sums_real(aTP_Lamm_iu),one_one_real)) ).

% power_half_series
tff(fact_6848_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2)))
      <=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% even_set_encode_iff
tff(fact_6849_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_gt_zero
tff(fact_6850_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X4))) ) ) ).

% lemma_tan_total
tff(fact_6851_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_6852_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_6853_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_6854_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ).

% tan_monotone
tff(fact_6855_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,X4) = Y )
      & ! [Y4: real] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X4 ) ) ) ).

% tan_total
tff(fact_6856_set__decode__def,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_iv,X)) ).

% set_decode_def
tff(fact_6857_mask__eq__sum__exp,axiom,
    ! [N2: 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))),N2)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,N2))) ).

% mask_eq_sum_exp
tff(fact_6858_mask__eq__sum__exp,axiom,
    ! [N2: 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))),N2)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,N2))) ).

% mask_eq_sum_exp
tff(fact_6859_zero__less__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(power_power_real(A),N2)))
    <=> ( ( N2 = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & ( A != zero_zero_real ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_6860_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N2)))
    <=> ( ( N2 = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & ( A != zero_zero_rat ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_6861_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N2)))
    <=> ( ( N2 = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & ( A != zero_zero_int ) )
        | ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).

% zero_less_power_eq
tff(fact_6862_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_6863_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M)),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_6864_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_6865_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),M)),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N2))) ) ) ).

% sum_gp_multiplied
tff(fact_6866_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups2906978787729119204at_rat(aTP_Lamm_iw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.in_pairs
tff(fact_6867_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups3539618377306564664at_int(aTP_Lamm_ix(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.in_pairs
tff(fact_6868_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_iy(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.in_pairs
tff(fact_6869_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_iz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% sum.in_pairs
tff(fact_6870_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_6871_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_6872_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_6873_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).

% tan_pos_pi2_le
tff(fact_6874_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real)) ) ) ).

% tan_less_zero
tff(fact_6875_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) ) ) ) ).

% tan_mono_le
tff(fact_6876_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
            <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_6877_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real)) ) ).

% tan_bound_pi2
tff(fact_6878_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( ( aa_real_real(tan_real,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_6879_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_6880_arctan,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_6881_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% even_mask_div_iff'
tff(fact_6882_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% even_mask_div_iff'
tff(fact_6883_power__le__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(A),N2)),zero_zero_real))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6884_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),zero_zero_rat))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6885_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),zero_zero_int))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
        & ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
          | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6886_mask__eq__sum__exp__nat,axiom,
    ! [N2: 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))),N2)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,N2))) ).

% mask_eq_sum_exp_nat
tff(fact_6887_gauss__sum__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(suc,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_6888_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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_6889_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_6890_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_6891_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_6892_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_6893_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_6894_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_6895_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ? [Z: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))))
          & ( aa_real_real(tan_real,Z) = X ) ) ) ).

% tan_total_pi4
tff(fact_6896_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% even_mask_div_iff
tff(fact_6897_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) = zero_zero_int )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ) ).

% even_mask_div_iff
tff(fact_6898_double__gauss__sum,axiom,
    ! [N2: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N2)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N2)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_6899_double__gauss__sum,axiom,
    ! [N2: 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),N2))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),one_one_complex)) ).

% double_gauss_sum
tff(fact_6900_double__gauss__sum,axiom,
    ! [N2: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)) ).

% double_gauss_sum
tff(fact_6901_double__gauss__sum,axiom,
    ! [N2: 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),N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int)) ).

% double_gauss_sum
tff(fact_6902_double__gauss__sum,axiom,
    ! [N2: 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),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat)) ).

% double_gauss_sum
tff(fact_6903_double__gauss__sum,axiom,
    ! [N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N2)),one_one_real)) ).

% double_gauss_sum
tff(fact_6904_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N2: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_ja(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N2)),one_on7984719198319812577d_enat)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),A)),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N2)),D))) ).

% double_arith_series
tff(fact_6905_double__arith__series,axiom,
    ! [A: complex,D: complex,N2: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_jb(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),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,N2)),D))) ).

% double_arith_series
tff(fact_6906_double__arith__series,axiom,
    ! [A: rat,D: rat,N2: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_jc(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),D))) ).

% double_arith_series
tff(fact_6907_double__arith__series,axiom,
    ! [A: int,D: int,N2: 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_jd(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = 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,N2)),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,N2)),D))) ).

% double_arith_series
tff(fact_6908_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: 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_je(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = 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,N2)),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,N2)),D))) ).

% double_arith_series
tff(fact_6909_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jf(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N2)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),D))) ).

% double_arith_series
tff(fact_6910_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jg(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = 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,N2)),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,N2),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_6911_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_6912_Bernoulli__inequality__even,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),N2))) ) ).

% Bernoulli_inequality_even
tff(fact_6913_gauss__sum,axiom,
    ! [N2: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6914_gauss__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = 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,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_6915_arith__series,axiom,
    ! [A: int,D: int,N2: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_jd(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = 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,N2)),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,N2)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6916_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_je(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = 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,N2)),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,N2)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_6917_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2))) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N2)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,N2)),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6918_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: 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)),N2))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6919_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6920_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: 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)),N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6921_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: 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)),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6922_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N2)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_6923_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) = zero_zero_nat )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6924_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) = zero_zero_int )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
          & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(divide_divide_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6925_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),one_one_complex) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N2))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6926_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6927_sum__gp__offset,axiom,
    ! [X: real,M: nat,N2: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N2)),one_one_real) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N2))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_offset
tff(fact_6928_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ).

% tan_half
tff(fact_6929_tan__half,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),one_one_complex)) ).

% tan_half
tff(fact_6930_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6931_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2)) = 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,N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_6932_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_6933_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_6934_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_6935_lemma__termdiff2,axiom,
    ! [H: complex,Z2: complex,N2: 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(Z2),H)),N2)),aa_nat_complex(power_power_complex(Z2),N2))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),aa_nat_complex(power_power_complex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ji(H),Z2),N2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6936_lemma__termdiff2,axiom,
    ! [H: rat,Z2: rat,N2: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z2),H)),N2)),aa_nat_rat(power_power_rat(Z2),N2))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(power_power_rat(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jk(H),Z2),N2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6937_lemma__termdiff2,axiom,
    ! [H: real,Z2: real,N2: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),H)),N2)),aa_nat_real(power_power_real(Z2),N2))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jm(H),Z2),N2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_6938_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_jn(C),X))
     => pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jo(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_jn(C),X)))) ) ).

% diffs_equiv
tff(fact_6939_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_jp(C),X))
     => pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_jq(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_jp(C),X)))) ) ).

% diffs_equiv
tff(fact_6940_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                    & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                     => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_6941_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_6942_sin__paired,axiom,
    ! [X: real] : pp(aa_real_bool(sums_real(aTP_Lamm_jr(X)),aa_real_real(sin_real,X))) ).

% sin_paired
tff(fact_6943_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_6944_intind,axiom,
    ! [I: nat,N2: nat,P: fun_nat_bool,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N2))
     => ( pp(aa_nat_bool(P,X))
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N2,X)),I))) ) ) ).

% intind
tff(fact_6945_intind,axiom,
    ! [I: nat,N2: nat,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N2))
     => ( pp(aa_VEBT_VEBT_bool(P,X))
       => pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N2,X),I))) ) ) ).

% intind
tff(fact_6946_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_6947_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,X) = aa_nat_set_nat(set_ord_lessThan_nat,Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_6948_lessThan__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( set_or5984915006950818249n_real(X) = set_or5984915006950818249n_real(Y) )
    <=> ( X = Y ) ) ).

% lessThan_eq_iff
tff(fact_6949_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or890127255671739683et_nat(K))
    <=> pp(aa_set_nat_bool(ord_less_set_nat(I),K)) ) ).

% lessThan_iff
tff(fact_6950_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( member_rat(I,set_ord_lessThan_rat(K))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),K)) ) ).

% lessThan_iff
tff(fact_6951_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_lessThan_num(K))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),K)) ) ).

% lessThan_iff
tff(fact_6952_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_lessThan_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) ) ).

% lessThan_iff
tff(fact_6953_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).

% lessThan_iff
tff(fact_6954_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( member_real(I,set_or5984915006950818249n_real(K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).

% lessThan_iff
tff(fact_6955_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_6956_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_6957_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_6958_int__dvd__int__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N2)) ) ).

% int_dvd_int_iff
tff(fact_6959_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,X) = replicate_VEBT_VEBT(N2,Y) )
    <=> ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_6960_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6961_finite__lessThan,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K))) ).

% finite_lessThan
tff(fact_6962_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% semiring_norm(80)
tff(fact_6963_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_lessThan_rat(X),set_ord_lessThan_rat(Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6964_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(X),set_ord_lessThan_num(Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6965_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6966_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6967_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ord_less_eq_set_real(set_or5984915006950818249n_real(X),set_or5984915006950818249n_real(Y))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).

% lessThan_subset_iff
tff(fact_6968_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_6969_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_6970_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_6971_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_6972_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_6973_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_6974_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_6975_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% real_sqrt_le_0_iff
tff(fact_6976_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ).

% real_sqrt_ge_0_iff
tff(fact_6977_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_6978_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_6979_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_6980_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_6981_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_6982_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_6983_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_6984_Ball__set__replicate,axiom,
    ! [N2: nat,A: nat,P: fun_nat_bool] :
      ( ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N2,A))))
         => pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6985_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P: fun_int_bool] :
      ( ! [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N2,A))))
         => pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6986_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N2,A))))
         => pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_6987_Bex__set__replicate,axiom,
    ! [N2: nat,A: nat,P: fun_nat_bool] :
      ( ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),set_nat2(replicate_nat(N2,A))))
          & pp(aa_nat_bool(P,X3)) )
    <=> ( pp(aa_nat_bool(P,A))
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6988_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P: fun_int_bool] :
      ( ? [X3: int] :
          ( pp(aa_set_int_bool(member_int(X3),set_int2(replicate_int(N2,A))))
          & pp(aa_int_bool(P,X3)) )
    <=> ( pp(aa_int_bool(P,A))
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6989_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
      ( ? [X3: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(replicate_VEBT_VEBT(N2,A))))
          & pp(aa_VEBT_VEBT_bool(P,X3)) )
    <=> ( pp(aa_VEBT_VEBT_bool(P,A))
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_6990_in__set__replicate,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( pp(aa_set_complex_bool(member_complex(X),set_complex2(replicate_complex(N2,Y))))
    <=> ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6991_in__set__replicate,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( member_real(X,set_real2(replicate_real(N2,Y)))
    <=> ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6992_in__set__replicate,axiom,
    ! [X: set_nat,N2: nat,Y: set_nat] :
      ( member_set_nat(X,set_set_nat2(replicate_set_nat(N2,Y)))
    <=> ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6993_in__set__replicate,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N2,Y))))
    <=> ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6994_in__set__replicate,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( pp(aa_set_int_bool(member_int(X),set_int2(replicate_int(N2,Y))))
    <=> ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6995_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N2,Y))))
    <=> ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_6996_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N2))
     => ( aa_nat_nat(nth_nat(replicate_nat(N2,X)),I) = X ) ) ).

% nth_replicate
tff(fact_6997_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: vEBT_VEBT] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N2))
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N2,X),I) = X ) ) ).

% nth_replicate
tff(fact_6998_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% semiring_norm(81)
tff(fact_6999_semiring__norm_I77_J,axiom,
    ! [N2: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit1,N2))) ).

% semiring_norm(77)
tff(fact_7000_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_7001_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7002_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7003_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7004_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_7005_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7006_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7007_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7008_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7009_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2))),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_Suc
tff(fact_7010_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% fact_Suc
tff(fact_7011_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))),aa_nat_real(semiri2265585572941072030t_real,N2)) ).

% fact_Suc
tff(fact_7012_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N2))),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) ).

% fact_Suc
tff(fact_7013_fact__Suc,axiom,
    ! [N2: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,N2)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N2))),aa_nat_complex(semiri5044797733671781792omplex,N2)) ).

% fact_Suc
tff(fact_7014_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_rat(G,N2)) ).

% sum.lessThan_Suc
tff(fact_7015_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_int(G,N2)) ).

% sum.lessThan_Suc
tff(fact_7016_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_nat(G,N2)) ).

% sum.lessThan_Suc
tff(fact_7017_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_real(G,N2)) ).

% sum.lessThan_Suc
tff(fact_7018_zdiv__numeral__Bit1,axiom,
    ! [V: num,W2: 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,W2))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W2)) ).

% zdiv_numeral_Bit1
tff(fact_7019_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2)) ) ).

% semiring_norm(79)
tff(fact_7020_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N2)))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% semiring_norm(74)
tff(fact_7021_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7022_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_7023_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_7024_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_7025_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N2)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N2)) ).

% div_Suc_eq_div_add3
tff(fact_7026_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_7027_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N2)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N2)) ).

% mod_Suc_eq_mod_add3
tff(fact_7028_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_7029_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_7030_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7031_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_7032_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7033_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_7034_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_nat_real(power_power_real(aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_7035_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(power_power_real(aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_7036_zmod__numeral__Bit1,axiom,
    ! [V: num,W2: 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,W2))) = 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,W2)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_7037_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7038_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N2)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N2),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_7039_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_ey(F),A2) ).

% int_sum
tff(fact_7040_int__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_fb(F),A2) ).

% int_sum
tff(fact_7041_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_7042_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,N2: real] :
      ( ! [X4: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(Q,X4)),aa_real_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(N2))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(N2))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_js(Q),P),set_or5984915006950818249n_real(N2)) ) ) ).

% sum_diff_distrib
tff(fact_7043_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,N2: nat] :
      ( ! [X4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X4)),aa_nat_nat(P,X4)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jt(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ) ) ).

% sum_diff_distrib
tff(fact_7044_lessThan__non__empty,axiom,
    ! [X: int] : set_ord_lessThan_int(X) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_7045_lessThan__non__empty,axiom,
    ! [X: real] : set_or5984915006950818249n_real(X) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_7046_infinite__Iio,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_lessThan_int(A))) ).

% infinite_Iio
tff(fact_7047_infinite__Iio,axiom,
    ! [A: real] : ~ finite_finite_real(set_or5984915006950818249n_real(A)) ).

% infinite_Iio
tff(fact_7048_fact__nonzero,axiom,
    ! [N2: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N2) != zero_zero_rat ).

% fact_nonzero
tff(fact_7049_fact__nonzero,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) != zero_zero_int ).

% fact_nonzero
tff(fact_7050_fact__nonzero,axiom,
    ! [N2: nat] : aa_nat_real(semiri2265585572941072030t_real,N2) != zero_zero_real ).

% fact_nonzero
tff(fact_7051_fact__nonzero,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) != zero_zero_nat ).

% fact_nonzero
tff(fact_7052_fact__nonzero,axiom,
    ! [N2: nat] : aa_nat_complex(semiri5044797733671781792omplex,N2) != zero_zero_complex ).

% fact_nonzero
tff(fact_7053_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_7054_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_7055_uminus__dvd__conv_I2_J,axiom,
    ! [D: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(uminus_uminus_int,T))) ) ).

% uminus_dvd_conv(2)
tff(fact_7056_uminus__dvd__conv_I1_J,axiom,
    ! [D: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,D)),T)) ) ).

% uminus_dvd_conv(1)
tff(fact_7057_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_7058_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
       => ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B) ) ) ) ).

% zdvd_antisym_abs
tff(fact_7059_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_ju(U)) ).

% lessThan_def
tff(fact_7060_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_jv,U)) ).

% lessThan_def
tff(fact_7061_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_jw,U)) ).

% lessThan_def
tff(fact_7062_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_jx,U)) ).

% lessThan_def
tff(fact_7063_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,U)) ).

% lessThan_def
tff(fact_7064_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_jy,U)) ).

% lessThan_def
tff(fact_7065_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_7066_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( aa_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_7067_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_7068_Iio__eq__empty__iff,axiom,
    ! [N2: extended_enat] :
      ( ( set_or8419480210114673929d_enat(N2) = bot_bo7653980558646680370d_enat )
    <=> ( N2 = bot_bo4199563552545308370d_enat ) ) ).

% Iio_eq_empty_iff
tff(fact_7069_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N2) = bot_bot_set_nat )
    <=> ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_7070_fact__ge__zero,axiom,
    ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2))) ).

% fact_ge_zero
tff(fact_7071_fact__ge__zero,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N2))) ).

% fact_ge_zero
tff(fact_7072_fact__ge__zero,axiom,
    ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N2))) ).

% fact_ge_zero
tff(fact_7073_fact__ge__zero,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ).

% fact_ge_zero
tff(fact_7074_fact__gt__zero,axiom,
    ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2))) ).

% fact_gt_zero
tff(fact_7075_fact__gt__zero,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N2))) ).

% fact_gt_zero
tff(fact_7076_fact__gt__zero,axiom,
    ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N2))) ).

% fact_gt_zero
tff(fact_7077_fact__gt__zero,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ).

% fact_gt_zero
tff(fact_7078_fact__not__neg,axiom,
    ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),zero_zero_rat)) ).

% fact_not_neg
tff(fact_7079_fact__not__neg,axiom,
    ! [N2: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),zero_zero_int)) ).

% fact_not_neg
tff(fact_7080_fact__not__neg,axiom,
    ! [N2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,N2)),zero_zero_real)) ).

% fact_not_neg
tff(fact_7081_fact__not__neg,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),zero_zero_nat)) ).

% fact_not_neg
tff(fact_7082_fact__ge__1,axiom,
    ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N2))) ).

% fact_ge_1
tff(fact_7083_fact__ge__1,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N2))) ).

% fact_ge_1
tff(fact_7084_fact__ge__1,axiom,
    ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,N2))) ).

% fact_ge_1
tff(fact_7085_fact__ge__1,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ).

% fact_ge_1
tff(fact_7086_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_7087_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(M),set_ord_lessThan_rat(N2))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M),N2)) ) ).

% lessThan_strict_subset_iff
tff(fact_7088_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( ord_less_set_num(set_ord_lessThan_num(M),set_ord_lessThan_num(N2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2)) ) ).

% lessThan_strict_subset_iff
tff(fact_7089_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N2)) ) ).

% lessThan_strict_subset_iff
tff(fact_7090_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% lessThan_strict_subset_iff
tff(fact_7091_lessThan__strict__subset__iff,axiom,
    ! [M: real,N2: real] :
      ( ord_less_set_real(set_or5984915006950818249n_real(M),set_or5984915006950818249n_real(N2))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M),N2)) ) ).

% lessThan_strict_subset_iff
tff(fact_7092_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N2))) ) ).

% fact_mono
tff(fact_7093_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N2))) ) ).

% fact_mono
tff(fact_7094_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N2))) ) ).

% fact_mono
tff(fact_7095_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ) ).

% fact_mono
tff(fact_7096_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),set_int2(Xs)))
         => ( X4 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7097_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(Xs)))
         => ( X4 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7098_replicate__length__same,axiom,
    ! [Xs: list_bool,X: bool] :
      ( ! [X4: bool] :
          ( member_bool(X4,set_bool2(Xs))
         => ( pp(X4)
          <=> pp(X) ) )
     => ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7099_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),set_nat2(Xs)))
         => ( X4 = X ) )
     => ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_7100_replicate__eqI,axiom,
    ! [Xs: list_complex,N2: nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = N2 )
     => ( ! [Y3: complex] :
            ( pp(aa_set_complex_bool(member_complex(Y3),set_complex2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_complex(N2,X) ) ) ) ).

% replicate_eqI
tff(fact_7101_replicate__eqI,axiom,
    ! [Xs: list_real,N2: nat,X: real] :
      ( ( size_size_list_real(Xs) = N2 )
     => ( ! [Y3: real] :
            ( member_real(Y3,set_real2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_real(N2,X) ) ) ) ).

% replicate_eqI
tff(fact_7102_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N2: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = N2 )
     => ( ! [Y3: set_nat] :
            ( member_set_nat(Y3,set_set_nat2(Xs))
           => ( Y3 = X ) )
       => ( Xs = replicate_set_nat(N2,X) ) ) ) ).

% replicate_eqI
tff(fact_7103_replicate__eqI,axiom,
    ! [Xs: list_int,N2: nat,X: int] :
      ( ( size_size_list_int(Xs) = N2 )
     => ( ! [Y3: int] :
            ( pp(aa_set_int_bool(member_int(Y3),set_int2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_int(N2,X) ) ) ) ).

% replicate_eqI
tff(fact_7104_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N2 )
     => ( ! [Y3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),set_VEBT_VEBT2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_VEBT_VEBT(N2,X) ) ) ) ).

% replicate_eqI
tff(fact_7105_replicate__eqI,axiom,
    ! [Xs: list_bool,N2: nat,X: bool] :
      ( ( size_size_list_bool(Xs) = N2 )
     => ( ! [Y3: bool] :
            ( member_bool(Y3,set_bool2(Xs))
           => ( pp(Y3)
            <=> pp(X) ) )
       => ( Xs = replicate_bool(N2,X) ) ) ) ).

% replicate_eqI
tff(fact_7106_replicate__eqI,axiom,
    ! [Xs: list_nat,N2: nat,X: nat] :
      ( ( size_size_list_nat(Xs) = N2 )
     => ( ! [Y3: nat] :
            ( pp(aa_set_nat_bool(member_nat(Y3),set_nat2(Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate_nat(N2,X) ) ) ) ).

% replicate_eqI
tff(fact_7107_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,M))) ) ).

% fact_dvd
tff(fact_7108_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,N2)),aa_nat_real(semiri2265585572941072030t_real,M))) ) ).

% fact_dvd
tff(fact_7109_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,M))) ) ).

% fact_dvd
tff(fact_7110_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N2),M))
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_7111_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N2))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N2),M)) ) ) ).

% zdvd_not_zless
tff(fact_7112_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),T))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),T))) ) ) ).

% zdvd_mono
tff(fact_7113_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N2)))
     => ( ( K != zero_zero_int )
       => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),N2)) ) ) ).

% zdvd_mult_cancel
tff(fact_7114_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N2) = bot_bot_set_nat )
    <=> ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_7115_zdvd__reduce,axiom,
    ! [K: int,N2: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,N2),aa_int_int(aa_int_fun_int_int(times_times_int,K),M))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N2)) ) ).

% zdvd_reduce
tff(fact_7116_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T))) ) ) ).

% zdvd_period
tff(fact_7117_abs__div,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y),X))
     => ( aa_int_int(abs_abs_int,aa_int_int(divide_divide_int(X),Y)) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_7118_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(G,X4)),aa_complex_nat(F,X4))) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_jz(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7119_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X4)),aa_real_nat(F,X4))) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_js(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7120_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(G,X4)),aa_set_nat_nat(F,X4))) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ka(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A2)),groups8294997508430121362at_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7121_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X4)),aa_int_nat(F,X4))) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kb(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7122_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G,X4)),aa_nat_nat(F,X4))) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jt(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_7123_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_gx,I)))) ) ).

% finite_divisors_int
tff(fact_7124_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7125_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7126_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ( groups977919841031483927at_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
            & ( aa_Pro4243173770044397973at_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7127_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2027974829824023292at_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
            & ( aa_Extended_enat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_Extended_enat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7128_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7129_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,N2: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,N2) )
     => ? [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) ) ) ).

% sum_SucD
tff(fact_7130_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa4: int] :
                ( pp(aa_set_int_bool(member_int(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_int_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7131_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa4: complex] :
                ( pp(aa_set_complex_bool(member_complex(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_complex_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7132_sum__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ( groups977919841031483927at_nat(F,A2) = one_one_nat )
      <=> ? [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
            & ( aa_Pro4243173770044397973at_nat(F,X3) = one_one_nat )
            & ! [Xa4: product_prod_nat_nat] :
                ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_Pro4243173770044397973at_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7133_sum__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2027974829824023292at_nat(F,A2) = one_one_nat )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
            & ( aa_Extended_enat_nat(F,X3) = one_one_nat )
            & ! [Xa4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_Extended_enat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7134_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa4: nat] :
                ( pp(aa_set_nat_bool(member_nat(Xa4),A2))
               => ( ( X3 != Xa4 )
                 => ( aa_nat_nat(F,Xa4) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7135_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_7136_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_7137_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: fun_na2180513548729900188l_real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun4674562053777771865t_real(aTP_Lamm_kc(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_literal_real(aa_nat8303615512603269253l_real(Diff,zero_zero_nat),zero_zero_literal) ) ) ) ).

% Maclaurin_zero
tff(fact_7138_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kd(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_7139_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: fun_nat_fun_rat_real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_ke(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_7140_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: fun_nat_fun_nat_real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_kf(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_7141_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: fun_nat_fun_int_real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_kg(X),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_7142_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ? [B8: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_kh(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(H),N2)),aa_nat_real(semiri2265585572941072030t_real,N2)))) ) ).

% Maclaurin_lemma
tff(fact_7143_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N2)),aa_num_rat(numeral_numeral_rat,N2))),one_one_rat) ).

% numeral_Bit1
tff(fact_7144_numeral__Bit1,axiom,
    ! [N2: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N2)),numera1916890842035813515d_enat(N2))),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_7145_numeral__Bit1,axiom,
    ! [N2: num] : numera6690914467698888265omplex(aa_num_num(bit1,N2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N2)),numera6690914467698888265omplex(N2))),one_one_complex) ).

% numeral_Bit1
tff(fact_7146_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N2)),aa_num_real(numeral_numeral_real,N2))),one_one_real) ).

% numeral_Bit1
tff(fact_7147_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2)) = 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,N2)),aa_num_nat(numeral_numeral_nat,N2))),one_one_nat) ).

% numeral_Bit1
tff(fact_7148_numeral__Bit1,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)) = 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,N2)),aa_num_int(numeral_numeral_int,N2))),one_one_int) ).

% numeral_Bit1
tff(fact_7149_eval__nat__numeral_I3_J,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2))) ).

% eval_nat_numeral(3)
tff(fact_7150_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N2))) ) ) ).

% fact_less_mono
tff(fact_7151_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N2))) ) ) ).

% fact_less_mono
tff(fact_7152_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N2))) ) ) ).

% fact_less_mono
tff(fact_7153_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ) ) ).

% fact_less_mono
tff(fact_7154_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q4: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(10)
tff(fact_7155_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q4: num,N2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(10)
tff(fact_7156_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q4: num,N2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(10)
tff(fact_7157_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q4: num,N2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(12)
tff(fact_7158_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q4: num,N2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(12)
tff(fact_7159_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q4: num,N2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) ).

% cong_exp_iff_simps(12)
tff(fact_7160_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,Q4)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_7161_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q4)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,Q4)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_7162_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q4: num,N2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q4)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N2),numera6620942414471956472nteger(Q4)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_7163_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_7164_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,N2),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_7165_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_7166_fact__le__power,axiom,
    ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2)))) ).

% fact_le_power
tff(fact_7167_fact__le__power,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N2)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2)))) ).

% fact_le_power
tff(fact_7168_fact__le__power,axiom,
    ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,N2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2)))) ).

% fact_le_power
tff(fact_7169_fact__le__power,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),N2)))) ).

% fact_le_power
tff(fact_7170_zdvd__imp__le,axiom,
    ! [Z2: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z2),N2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),N2)) ) ) ).

% zdvd_imp_le
tff(fact_7171_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),I))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I))) ) ) ).

% dvd_imp_le_int
tff(fact_7172_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ki(G),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nat_diff_reindex
tff(fact_7173_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_kj(G),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nat_diff_reindex
tff(fact_7174_numeral__code_I3_J,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,N2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,N2)),aa_num_rat(numeral_numeral_rat,N2))),one_one_rat) ).

% numeral_code(3)
tff(fact_7175_numeral__code_I3_J,axiom,
    ! [N2: num] : numera1916890842035813515d_enat(aa_num_num(bit1,N2)) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(N2)),numera1916890842035813515d_enat(N2))),one_on7984719198319812577d_enat) ).

% numeral_code(3)
tff(fact_7176_numeral__code_I3_J,axiom,
    ! [N2: num] : numera6690914467698888265omplex(aa_num_num(bit1,N2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N2)),numera6690914467698888265omplex(N2))),one_one_complex) ).

% numeral_code(3)
tff(fact_7177_numeral__code_I3_J,axiom,
    ! [N2: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,N2)),aa_num_real(numeral_numeral_real,N2))),one_one_real) ).

% numeral_code(3)
tff(fact_7178_numeral__code_I3_J,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2)) = 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,N2)),aa_num_nat(numeral_numeral_nat,N2))),one_one_nat) ).

% numeral_code(3)
tff(fact_7179_numeral__code_I3_J,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)) = 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,N2)),aa_num_int(numeral_numeral_int,N2))),one_one_int) ).

% numeral_code(3)
tff(fact_7180_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N2))
     => ( groups7754918857620584856omplex(aTP_Lamm_ar,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_bo(N2),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_7181_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N2))
     => ( groups7754918857620584856omplex(aTP_Lamm_ar,collect_complex(aTP_Lamm_bq(N2))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_7182_sqrt2__less__2,axiom,
    pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% sqrt2_less_2
tff(fact_7183_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_7184_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_7185_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_7186_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_7187_Suc3__eq__add__3,axiom,
    ! [N2: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N2) ).

% Suc3_eq_add_3
tff(fact_7188_suminf__le__const,axiom,
    ! [F: fun_nat_int,X: int] :
      ( summable_int(F)
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),X)) ) ) ).

% suminf_le_const
tff(fact_7189_suminf__le__const,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),X)) ) ) ).

% suminf_le_const
tff(fact_7190_suminf__le__const,axiom,
    ! [F: fun_nat_real,X: real] :
      ( summable_real(F)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),X)) ) ) ).

% suminf_le_const
tff(fact_7191_diffs__def,axiom,
    ! [C: fun_nat_complex,X2: nat] : aa_nat_complex(diffs_complex(C),X2) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X2))),aa_nat_complex(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_7192_diffs__def,axiom,
    ! [C: fun_nat_real,X2: nat] : aa_nat_real(diffs_real(C),X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X2))),aa_nat_real(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_7193_diffs__def,axiom,
    ! [C: fun_nat_rat,X2: nat] : aa_nat_rat(diffs_rat(C),X2) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X2))),aa_nat_rat(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_7194_diffs__def,axiom,
    ! [C: fun_nat_int,X2: nat] : aa_nat_int(diffs_int(C),X2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X2))),aa_nat_int(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_7195_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_7196_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_7197_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = 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_hk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_7198_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.lessThan_Suc_shift
tff(fact_7199_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_kk(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7200_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_kl(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7201_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_km(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_7202_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_ib(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7203_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_ic(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7204_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_id(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7205_zdvd__mult__cancel1,axiom,
    ! [M: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N2)),M))
      <=> ( aa_int_int(abs_abs_int,N2) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7206_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,X: int] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)))
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),X))
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7207_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,X: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)))
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X))
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7208_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,X: real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)))
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),X))
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7209_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( ( L = zero_zero_int )
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).

% mod_int_pos_iff
tff(fact_7210_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.atLeast1_atMost_eq
tff(fact_7211_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.atLeast1_atMost_eq
tff(fact_7212_sums__iff__shift,axiom,
    ! [F: fun_nat_real,N2: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),N2)),S))
    <=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))))) ) ).

% sums_iff_shift
tff(fact_7213_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,N2: nat,S: real] :
      ( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),N2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2)))))
    <=> pp(aa_real_bool(sums_real(F),S)) ) ).

% sums_iff_shift'
tff(fact_7214_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S: real,N2: nat] :
      ( pp(aa_real_bool(sums_real(F),S))
     => pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),N2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))))) ) ).

% sums_split_initial_segment
tff(fact_7215_bset_I9_J,axiom,
    ! [D: int,D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(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,X2),D4)),T))) ) ) ) ).

% bset(9)
tff(fact_7216_bset_I10_J,axiom,
    ! [D: int,D4: int,B2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),B2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(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,X2),D4)),T))) ) ) ) ).

% bset(10)
tff(fact_7217_aset_I9_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(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,X2),D4)),T))) ) ) ) ).

% aset(9)
tff(fact_7218_aset_I10_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa_set_int_bool(member_int(Xa3),set_or1266510415728281911st_int(one_one_int,D4)))
             => ! [Xb2: int] :
                  ( pp(aa_set_int_bool(member_int(Xb2),A2))
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(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,X2),D4)),T))) ) ) ) ).

% aset(10)
tff(fact_7219_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_7220_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_7221_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N2)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7222_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,N2)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7223_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N2)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N2),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7224_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7225_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7226_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q4))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7227_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N2) ).

% Suc_div_eq_add3_div
tff(fact_7228_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N2) ).

% Suc_mod_eq_add3_mod
tff(fact_7229_sum__count__set,axiom,
    ! [Xs: list_int,X5: set_int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),X5))
     => ( pp(aa_set_int_bool(finite_finite_int,X5))
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X5) = size_size_list_int(Xs) ) ) ) ).

% sum_count_set
tff(fact_7230_sum__count__set,axiom,
    ! [Xs: list_complex,X5: set_complex] :
      ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),X5))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,X5))
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X5) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_7231_sum__count__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),X5))
     => ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X5))
       => ( groups977919841031483927at_nat(count_4203492906077236349at_nat(Xs),X5) = size_s5460976970255530739at_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_7232_sum__count__set,axiom,
    ! [Xs: list_Extended_enat,X5: set_Extended_enat] :
      ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Xs)),X5))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,X5))
       => ( groups2027974829824023292at_nat(count_101369445342291426d_enat(Xs),X5) = size_s3941691890525107288d_enat(Xs) ) ) ) ).

% sum_count_set
tff(fact_7233_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X5: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X5)
     => ( finite5795047828879050333T_VEBT(X5)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X5) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_7234_sum__count__set,axiom,
    ! [Xs: list_bool,X5: set_bool] :
      ( ord_less_eq_set_bool(set_bool2(Xs),X5)
     => ( finite_finite_bool(X5)
       => ( groups5768501273728744877ol_nat(count_list_bool(Xs),X5) = size_size_list_bool(Xs) ) ) ) ).

% sum_count_set
tff(fact_7235_sum__count__set,axiom,
    ! [Xs: list_nat,X5: set_nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),X5))
     => ( pp(aa_set_nat_bool(finite_finite_nat,X5))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X5) = size_size_list_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_7236_one__diff__power__eq,axiom,
    ! [X: complex,N2: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N2)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq
tff(fact_7237_one__diff__power__eq,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N2)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq
tff(fact_7238_one__diff__power__eq,axiom,
    ! [X: int,N2: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq
tff(fact_7239_one__diff__power__eq,axiom,
    ! [X: real,N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq
tff(fact_7240_power__diff__1__eq,axiom,
    ! [X: complex,N2: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N2)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_1_eq
tff(fact_7241_power__diff__1__eq,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N2)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_1_eq
tff(fact_7242_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N2)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_1_eq
tff(fact_7243_power__diff__1__eq,axiom,
    ! [X: real,N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),N2)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_1_eq
tff(fact_7244_geometric__sum,axiom,
    ! [X: rat,N2: nat] :
      ( ( X != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N2)),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_7245_geometric__sum,axiom,
    ! [X: complex,N2: nat] :
      ( ( X != one_one_complex )
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N2)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_7246_geometric__sum,axiom,
    ! [X: real,N2: nat] :
      ( ( X != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),N2)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ) ).

% geometric_sum
tff(fact_7247_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_7248_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),K)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_7249_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_7250_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_7251_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_7252_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_7253_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_7254_sum__less__suminf,axiom,
    ! [F: fun_nat_int,N2: nat] :
      ( summable_int(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M5))) )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),suminf_int(F))) ) ) ).

% sum_less_suminf
tff(fact_7255_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,N2: nat] :
      ( summable_nat(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M5))) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),suminf_nat(F))) ) ) ).

% sum_less_suminf
tff(fact_7256_sum__less__suminf,axiom,
    ! [F: fun_nat_real,N2: nat] :
      ( summable_real(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M5))) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),suminf_real(F))) ) ) ).

% sum_less_suminf
tff(fact_7257_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = one_one_rat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_rat(semiri773545260158071498ct_rat,M) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7258_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = one_one_int ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_int(semiri1406184849735516958ct_int,M) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7259_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = one_one_real ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_real(semiri2265585572941072030t_real,M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7260_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = one_one_nat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7261_fact__num__eq__if,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = one_one_complex ) )
      & ( ( M != zero_zero_nat )
       => ( aa_nat_complex(semiri5044797733671781792omplex,M) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).

% fact_num_eq_if
tff(fact_7262_fact__reduce,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_rat(semiri773545260158071498ct_rat,N2) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7263_fact__reduce,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7264_fact__reduce,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_real(semiri2265585572941072030t_real,N2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7265_fact__reduce,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7266_fact__reduce,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_complex(semiri5044797733671781792omplex,N2) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7267_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_7268_sum__gp__strict,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_nat_complex(semiri8010041392384452111omplex,N2) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N2))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp_strict
tff(fact_7269_sum__gp__strict,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_nat_rat(semiri681578069525770553at_rat,N2) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N2))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp_strict
tff(fact_7270_sum__gp__strict,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_nat_real(semiri5074537144036343181t_real,N2) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),N2))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp_strict
tff(fact_7271_lemma__termdiff1,axiom,
    ! [Z2: complex,H: complex,M: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kn(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ko(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7272_lemma__termdiff1,axiom,
    ! [Z2: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kp(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kq(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7273_lemma__termdiff1,axiom,
    ! [Z2: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_kr(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ks(Z2),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7274_lemma__termdiff1,axiom,
    ! [Z2: real,H: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_kt(Z2),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ku(Z2),H),M)),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).

% lemma_termdiff1
tff(fact_7275_power__diff__sumr2,axiom,
    ! [X: complex,N2: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N2)),aa_nat_complex(power_power_complex(Y),N2)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_kv(X),N2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_7276_power__diff__sumr2,axiom,
    ! [X: rat,N2: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N2)),aa_nat_rat(power_power_rat(Y),N2)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_kw(X),N2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_7277_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N2)),aa_nat_int(power_power_int(Y),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_kx(X),N2),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_7278_power__diff__sumr2,axiom,
    ! [X: real,N2: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),N2)),aa_nat_real(power_power_real(Y),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ky(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% power_diff_sumr2
tff(fact_7279_diff__power__eq__sum,axiom,
    ! [X: complex,N2: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N2))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N2))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_kz(X),N2),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_7280_diff__power__eq__sum,axiom,
    ! [X: rat,N2: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_la(X),N2),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_7281_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),aa_nat_nat(suc,N2))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_lb(X),N2),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_7282_diff__power__eq__sum,axiom,
    ! [X: real,N2: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),aa_nat_nat(suc,N2))),aa_nat_real(power_power_real(Y),aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lc(X),N2),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)))) ).

% diff_power_eq_sum
tff(fact_7283_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_7284_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_7285_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_7286_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: fun_nat_rat,K5: rat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,P4)),K5)) )
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K5))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7287_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,P4)),K5)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K5))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N2)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7288_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,P4)),K5)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N2)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7289_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,P4)),K5)) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K5))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),K5))) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7290_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,N2: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M5))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),I))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),suminf_int(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7291_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,N2: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M5))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),I))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),suminf_nat(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7292_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,N2: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M5))) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),I))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),suminf_real(F))) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7293_one__diff__power__eq_H,axiom,
    ! [X: complex,N2: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N2)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ld(X),N2),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_7294_one__diff__power__eq_H,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),N2)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_le(X),N2),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_7295_one__diff__power__eq_H,axiom,
    ! [X: int,N2: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_lf(X),N2),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_7296_one__diff__power__eq_H,axiom,
    ! [X: real,N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lg(X),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% one_diff_power_eq'
tff(fact_7297_termdiff__converges,axiom,
    ! [X: real,K5: real,C: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K5))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X4)),K5))
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),X4)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_lh(X),C)) ) ) ).

% termdiff_converges
tff(fact_7298_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: fun_nat_complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K5))
     => ( ! [X4: complex] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X4)),K5))
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),X4)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_li(X),C)) ) ) ).

% termdiff_converges
tff(fact_7299_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).

% arsinh_real_aux
tff(fact_7300_real__sqrt__power__even,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(power_power_real(aa_real_real(sqrt,X)),N2) = aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_7301_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_7302_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_7303_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lj(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))),N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lk(F)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ll(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum_split_even_odd
tff(fact_7304_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_7305_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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_7306_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_7307_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X)))
     => ( aa_real_real(sin_real,X) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7308_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N2))
     => ( groups4538972089207619220nt_int(aTP_Lamm_av,set_or1266510415728281911st_int(M,N2)) = 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,N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,N2),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_7309_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)),one_one_nat))))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).

% sum_pos_lt_pair
tff(fact_7310_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))))
       => ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_7311_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(power_power_real(aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_7312_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),zero_zero_real))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lm(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N2))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_7313_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X))
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lm(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N2))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_7314_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X))
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ln(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N2))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_7315_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),X))
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ln(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N2))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_7316_sin__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(sin_coeff,X2) = zero_zero_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(sin_coeff,X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2)) ) ) ) ).

% sin_coeff_def
tff(fact_7317_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hk(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_7318_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cs(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).

% sum_bounds_lt_plus1
tff(fact_7319_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7320_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_7321_sumr__cos__zero__one,axiom,
    ! [N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lo),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_7322_fact__ge__self,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ).

% fact_ge_self
tff(fact_7323_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ) ).

% fact_mono_nat
tff(fact_7324_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ) ) ).

% fact_less_mono_nat
tff(fact_7325_sin__coeff__Suc,axiom,
    ! [N2: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,N2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))) ).

% sin_coeff_Suc
tff(fact_7326_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ).

% fact_ge_Suc_0_nat
tff(fact_7327_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ) ) ).

% dvd_fact
tff(fact_7328_cos__coeff__Suc,axiom,
    ! [N2: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N2))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))) ).

% cos_coeff_Suc
tff(fact_7329_fact__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(suc,M)))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2))) ) ) ).

% fact_diff_Suc
tff(fact_7330_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_7331_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N2))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)))) ).

% fact_fact_dvd_fact
tff(fact_7332_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,N2))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)))) ).

% fact_fact_dvd_fact
tff(fact_7333_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)))) ).

% fact_fact_dvd_fact
tff(fact_7334_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N2))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2)))) ).

% fact_fact_dvd_fact
tff(fact_7335_cos__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(cos_coeff,X2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2))
       => ( aa_nat_real(cos_coeff,X2) = zero_zero_real ) ) ) ).

% cos_coeff_def
tff(fact_7336_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N2))) ) ).

% choose_dvd
tff(fact_7337_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_nat_int(semiri1406184849735516958ct_int,N2))) ) ).

% choose_dvd
tff(fact_7338_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_nat_real(semiri2265585572941072030t_real,N2))) ) ).

% choose_dvd
tff(fact_7339_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N2))) ) ).

% choose_dvd
tff(fact_7340_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N2),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7341_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N2)) = unique5024387138958732305ep_int(aa_num_num(bit1,N2),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N2)))) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_7342_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2))
       => ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N2),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7343_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N2))
       => ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N2)) = unique5024387138958732305ep_int(aa_num_num(bit1,N2),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N2)))) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_7344_Maclaurin__exp__lt,axiom,
    ! [X: real,N2: nat] :
      ( ( X != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7345_monoseq__def,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,M6)),aa_nat_real(X5,N4))) )
        | ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N4)),aa_nat_real(X5,M6))) ) ) ) ).

% monoseq_def
tff(fact_7346_monoseq__def,axiom,
    ! [X5: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X5)
    <=> ( ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,M6)),aa_nat_set_nat(X5,N4))) )
        | ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N4)),aa_nat_set_nat(X5,M6))) ) ) ) ).

% monoseq_def
tff(fact_7347_monoseq__def,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,M6)),aa_nat_rat(X5,N4))) )
        | ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N4)),aa_nat_rat(X5,M6))) ) ) ) ).

% monoseq_def
tff(fact_7348_monoseq__def,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,M6)),aa_nat_num(X5,N4))) )
        | ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N4)),aa_nat_num(X5,M6))) ) ) ) ).

% monoseq_def
tff(fact_7349_monoseq__def,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,M6)),aa_nat_nat(X5,N4))) )
        | ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N4)),aa_nat_nat(X5,M6))) ) ) ) ).

% monoseq_def
tff(fact_7350_monoseq__def,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,M6)),aa_nat_int(X5,N4))) )
        | ! [M6: nat,N4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M6),N4))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N4)),aa_nat_int(X5,M6))) ) ) ) ).

% monoseq_def
tff(fact_7351_monoI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N)),aa_nat_real(X5,M5))) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI2
tff(fact_7352_monoI2,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N)),aa_nat_set_nat(X5,M5))) )
     => topolo7278393974255667507et_nat(X5) ) ).

% monoI2
tff(fact_7353_monoI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N)),aa_nat_rat(X5,M5))) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI2
tff(fact_7354_monoI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N)),aa_nat_num(X5,M5))) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI2
tff(fact_7355_monoI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N)),aa_nat_nat(X5,M5))) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI2
tff(fact_7356_monoI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N)),aa_nat_int(X5,M5))) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI2
tff(fact_7357_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))) ) ).

% exp_less_mono
tff(fact_7358_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel_iff
tff(fact_7359_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_7360_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_7361_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,X) = one_one_real )
    <=> ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_7362_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% exp_less_one_iff
tff(fact_7363_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% one_less_exp_iff
tff(fact_7364_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% exp_le_one_iff
tff(fact_7365_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% one_le_exp_iff
tff(fact_7366_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).

% exp_ln
tff(fact_7367_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% exp_ln_iff
tff(fact_7368_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)))
    <=> unique6319869463603278526ux_int(unique5052692396658037445od_int(N2,M)) ) ).

% dvd_numeral_simp
tff(fact_7369_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)))
    <=> pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(N2,M))) ) ).

% dvd_numeral_simp
tff(fact_7370_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_7371_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_7372_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N2)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_7373_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_7374_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N2)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_7375_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N2)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_7376_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% exp_less_cancel
tff(fact_7377_exp__not__eq__zero,axiom,
    ! [X: real] : aa_real_real(exp_real,X) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_7378_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_less_zero
tff(fact_7379_exp__gt__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_gt_zero
tff(fact_7380_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
     => ? [X4: real] : aa_real_real(exp_real,X4) = Y ) ).

% exp_total
tff(fact_7381_exp__ge__zero,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X))) ).

% exp_ge_zero
tff(fact_7382_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real)) ).

% not_exp_le_zero
tff(fact_7383_mult__exp__exp,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).

% mult_exp_exp
tff(fact_7384_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_7385_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X))) ) ).

% exp_gt_one
tff(fact_7386_exp__minus__inverse,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) = one_one_real ).

% exp_minus_inverse
tff(fact_7387_exp__minus__inverse,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X))) = one_one_complex ).

% exp_minus_inverse
tff(fact_7388_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_7389_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
     => ? [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real)))
          & ( aa_real_real(exp_real,X4) = Y ) ) ) ).

% lemma_exp_total
tff(fact_7390_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_7391_exp__divide__power__eq,axiom,
    ! [N2: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,N2)))),N2) = aa_complex_complex(exp_complex,X) ) ) ).

% exp_divide_power_eq
tff(fact_7392_exp__divide__power__eq,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_real(power_power_real(aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N2)))),N2) = aa_real_real(exp_real,X) ) ) ).

% exp_divide_power_eq
tff(fact_7393_tanh__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% tanh_altdef
tff(fact_7394_tanh__altdef,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% tanh_altdef
tff(fact_7395_divmod__int__def,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(M,N2) = product_Pair_int_int(aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))) ).

% divmod_int_def
tff(fact_7396_divmod__def,axiom,
    ! [M: num,N2: num] : unique3479559517661332726nteger(M,N2) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N2)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N2))) ).

% divmod_def
tff(fact_7397_divmod__def,axiom,
    ! [M: num,N2: num] : unique5052692396658037445od_int(M,N2) = product_Pair_int_int(aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N2))) ).

% divmod_def
tff(fact_7398_divmod__def,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(M,N2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ).

% divmod_def
tff(fact_7399_divmod_H__nat__def,axiom,
    ! [M: num,N2: num] : unique5055182867167087721od_nat(M,N2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ).

% divmod'_nat_def
tff(fact_7400_exp__bound,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% exp_bound
tff(fact_7401_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_7402_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N2))),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N2)))),N2)),aa_real_real(exp_real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7403_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N2)))),N2)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7404_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5055182867167087721od_nat(M,N2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5055182867167087721od_nat(M,N2) = unique5026877609467782581ep_nat(N2,unique5055182867167087721od_nat(M,aa_num_num(bit0,N2))) ) ) ) ).

% divmod_divmod_step
tff(fact_7405_divmod__divmod__step,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5052692396658037445od_int(M,N2) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N2))
       => ( unique5052692396658037445od_int(M,N2) = unique5024387138958732305ep_int(N2,unique5052692396658037445od_int(M,aa_num_num(bit0,N2))) ) ) ) ).

% divmod_divmod_step
tff(fact_7406_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7407_mono__SucI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N)),aa_nat_real(X5,aa_nat_nat(suc,N))))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI1
tff(fact_7408_mono__SucI1,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [N: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N)),aa_nat_set_nat(X5,aa_nat_nat(suc,N))))
     => topolo7278393974255667507et_nat(X5) ) ).

% mono_SucI1
tff(fact_7409_mono__SucI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N)),aa_nat_rat(X5,aa_nat_nat(suc,N))))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI1
tff(fact_7410_mono__SucI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N)),aa_nat_num(X5,aa_nat_nat(suc,N))))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI1
tff(fact_7411_mono__SucI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N)),aa_nat_nat(X5,aa_nat_nat(suc,N))))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI1
tff(fact_7412_mono__SucI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N)),aa_nat_int(X5,aa_nat_nat(suc,N))))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI1
tff(fact_7413_mono__SucI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N))),aa_nat_real(X5,N)))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI2
tff(fact_7414_mono__SucI2,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [N: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,aa_nat_nat(suc,N))),aa_nat_set_nat(X5,N)))
     => topolo7278393974255667507et_nat(X5) ) ).

% mono_SucI2
tff(fact_7415_mono__SucI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N))),aa_nat_rat(X5,N)))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI2
tff(fact_7416_mono__SucI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N))),aa_nat_num(X5,N)))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI2
tff(fact_7417_mono__SucI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N))),aa_nat_nat(X5,N)))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI2
tff(fact_7418_mono__SucI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N))),aa_nat_int(X5,N)))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI2
tff(fact_7419_monoseq__Suc,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [N4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N4)),aa_nat_real(X5,aa_nat_nat(suc,N4))))
        | ! [N4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N4))),aa_nat_real(X5,N4))) ) ) ).

% monoseq_Suc
tff(fact_7420_monoseq__Suc,axiom,
    ! [X5: fun_nat_set_nat] :
      ( topolo7278393974255667507et_nat(X5)
    <=> ( ! [N4: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N4)),aa_nat_set_nat(X5,aa_nat_nat(suc,N4))))
        | ! [N4: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,aa_nat_nat(suc,N4))),aa_nat_set_nat(X5,N4))) ) ) ).

% monoseq_Suc
tff(fact_7421_monoseq__Suc,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [N4: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N4)),aa_nat_rat(X5,aa_nat_nat(suc,N4))))
        | ! [N4: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N4))),aa_nat_rat(X5,N4))) ) ) ).

% monoseq_Suc
tff(fact_7422_monoseq__Suc,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [N4: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N4)),aa_nat_num(X5,aa_nat_nat(suc,N4))))
        | ! [N4: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N4))),aa_nat_num(X5,N4))) ) ) ).

% monoseq_Suc
tff(fact_7423_monoseq__Suc,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [N4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N4)),aa_nat_nat(X5,aa_nat_nat(suc,N4))))
        | ! [N4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N4))),aa_nat_nat(X5,N4))) ) ) ).

% monoseq_Suc
tff(fact_7424_monoseq__Suc,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [N4: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N4)),aa_nat_int(X5,aa_nat_nat(suc,N4))))
        | ! [N4: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N4))),aa_nat_int(X5,N4))) ) ) ).

% monoseq_Suc
tff(fact_7425_monoI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,M5)),aa_nat_real(X5,N))) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI1
tff(fact_7426_monoI1,axiom,
    ! [X5: fun_nat_set_nat] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,M5)),aa_nat_set_nat(X5,N))) )
     => topolo7278393974255667507et_nat(X5) ) ).

% monoI1
tff(fact_7427_monoI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,M5)),aa_nat_rat(X5,N))) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI1
tff(fact_7428_monoI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,M5)),aa_nat_num(X5,N))) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI1
tff(fact_7429_monoI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,M5)),aa_nat_nat(X5,N))) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI1
tff(fact_7430_monoI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M5: nat,N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,M5)),aa_nat_int(X5,N))) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI1
tff(fact_7431_minus__one__div__numeral,axiom,
    ! [N2: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N2)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N2))) ).

% minus_one_div_numeral
tff(fact_7432_one__div__minus__numeral,axiom,
    ! [N2: num] : aa_int_int(divide_divide_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N2))) ).

% one_div_minus_numeral
tff(fact_7433_minus__numeral__div__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N2))) ).

% minus_numeral_div_numeral
tff(fact_7434_numeral__div__minus__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,N2))) ).

% numeral_div_minus_numeral
tff(fact_7435_pochhammer__double,axiom,
    ! [Z2: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = 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))),N2)))),aa_nat_complex(comm_s2602460028002588243omplex(Z2),N2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),N2)) ).

% pochhammer_double
tff(fact_7436_pochhammer__double,axiom,
    ! [Z2: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = 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))),N2)))),aa_nat_real(comm_s7457072308508201937r_real(Z2),N2))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),N2)) ).

% pochhammer_double
tff(fact_7437_pochhammer__double,axiom,
    ! [Z2: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),N2)) ).

% pochhammer_double
tff(fact_7438_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_7439_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(comm_s7457072308508201937r_real(A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_7440_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_7441_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_7442_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_7443_pochhammer__pos,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),N2))) ) ).

% pochhammer_pos
tff(fact_7444_pochhammer__pos,axiom,
    ! [X: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N2))) ) ).

% pochhammer_pos
tff(fact_7445_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N2))) ) ).

% pochhammer_pos
tff(fact_7446_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N2))) ) ).

% pochhammer_pos
tff(fact_7447_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),M) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N2) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7448_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7449_pochhammer__eq__0__mono,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N2) = zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7450_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = zero_zero_rat )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7451_pochhammer__fact,axiom,
    ! [N2: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N2) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),N2) ).

% pochhammer_fact
tff(fact_7452_pochhammer__fact,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),N2) ).

% pochhammer_fact
tff(fact_7453_pochhammer__fact,axiom,
    ! [N2: nat] : aa_nat_real(semiri2265585572941072030t_real,N2) = aa_nat_real(comm_s7457072308508201937r_real(one_one_real),N2) ).

% pochhammer_fact
tff(fact_7454_pochhammer__fact,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),N2) ).

% pochhammer_fact
tff(fact_7455_pochhammer__fact,axiom,
    ! [N2: nat] : aa_nat_complex(semiri5044797733671781792omplex,N2) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),N2) ).

% pochhammer_fact
tff(fact_7456_pochhammer__nonneg,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(comm_s7457072308508201937r_real(X),N2))) ) ).

% pochhammer_nonneg
tff(fact_7457_pochhammer__nonneg,axiom,
    ! [X: rat,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N2))) ) ).

% pochhammer_nonneg
tff(fact_7458_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N2))) ) ).

% pochhammer_nonneg
tff(fact_7459_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N2))) ) ).

% pochhammer_nonneg
tff(fact_7460_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N2) = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N2) = zero_zero_complex ) ) ) ).

% pochhammer_0_left
tff(fact_7461_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N2) = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(zero_zero_real),N2) = zero_zero_real ) ) ) ).

% pochhammer_0_left
tff(fact_7462_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N2) = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N2) = zero_zero_rat ) ) ) ).

% pochhammer_0_left
tff(fact_7463_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N2) = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N2) = zero_zero_nat ) ) ) ).

% pochhammer_0_left
tff(fact_7464_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N2) = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N2) = zero_zero_int ) ) ) ).

% pochhammer_0_left
tff(fact_7465_pochhammer__rec,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N2)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),N2)) ).

% pochhammer_rec
tff(fact_7466_pochhammer__rec,axiom,
    ! [A: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),N2)) ).

% pochhammer_rec
tff(fact_7467_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),N2)) ).

% pochhammer_rec
tff(fact_7468_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_7469_pochhammer__rec,axiom,
    ! [A: int,N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N2)) = 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)),N2)) ).

% pochhammer_rec
tff(fact_7470_pochhammer__Suc,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N2)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),N2)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N2))) ).

% pochhammer_Suc
tff(fact_7471_pochhammer__Suc,axiom,
    ! [A: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(A),N2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N2))) ).

% pochhammer_Suc
tff(fact_7472_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),N2)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).

% pochhammer_Suc
tff(fact_7473_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N2))) ).

% pochhammer_Suc
tff(fact_7474_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N2))) ).

% pochhammer_Suc
tff(fact_7475_pochhammer__rec_H,axiom,
    ! [Z2: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z2),aa_nat_nat(suc,N2)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z2),aa_nat_complex(semiri8010041392384452111omplex,N2))),aa_nat_complex(comm_s2602460028002588243omplex(Z2),N2)) ).

% pochhammer_rec'
tff(fact_7476_pochhammer__rec_H,axiom,
    ! [Z2: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z2),aa_nat_nat(suc,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),aa_nat_real(semiri5074537144036343181t_real,N2))),aa_nat_real(comm_s7457072308508201937r_real(Z2),N2)) ).

% pochhammer_rec'
tff(fact_7477_pochhammer__rec_H,axiom,
    ! [Z2: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(suc,N2)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z2),aa_nat_rat(semiri681578069525770553at_rat,N2))),aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N2)) ).

% pochhammer_rec'
tff(fact_7478_pochhammer__rec_H,axiom,
    ! [Z2: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,N2))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N2)) ).

% pochhammer_rec'
tff(fact_7479_pochhammer__rec_H,axiom,
    ! [Z2: int,N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z2),aa_nat_nat(suc,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,N2))),aa_nat_int(comm_s4660882817536571857er_int(Z2),N2)) ).

% pochhammer_rec'
tff(fact_7480_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N2) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N2))
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7481_pochhammer__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(A),N2) = zero_zero_real )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N2))
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7482_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N2))
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7483_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K) = zero_z3403309356797280102nteger )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7484_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N2))),K) = zero_zero_complex )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7485_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N2))),K) = zero_zero_real )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7486_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K) = zero_zero_rat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7487_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K) = zero_zero_int )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K)) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7488_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7489_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N2))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7490_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N2))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7491_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7492_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7493_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7494_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N2))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7495_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N2))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7496_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7497_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7498_pochhammer__product_H,axiom,
    ! [Z2: complex,N2: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z2),N2)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_nat_complex(semiri8010041392384452111omplex,N2))),M)) ).

% pochhammer_product'
tff(fact_7499_pochhammer__product_H,axiom,
    ! [Z2: real,N2: nat,M: nat] : aa_nat_real(comm_s7457072308508201937r_real(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z2),N2)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),aa_nat_real(semiri5074537144036343181t_real,N2))),M)) ).

% pochhammer_product'
tff(fact_7500_pochhammer__product_H,axiom,
    ! [Z2: rat,N2: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z2),aa_nat_rat(semiri681578069525770553at_rat,N2))),M)) ).

% pochhammer_product'
tff(fact_7501_pochhammer__product_H,axiom,
    ! [Z2: nat,N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N2)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,N2))),M)) ).

% pochhammer_product'
tff(fact_7502_pochhammer__product_H,axiom,
    ! [Z2: int,N2: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z2),N2)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,N2))),M)) ).

% pochhammer_product'
tff(fact_7503_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Z2),N2) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z2),M)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_nat_complex(semiri8010041392384452111omplex,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_7504_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_real(comm_s7457072308508201937r_real(Z2),N2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z2),M)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_7505_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z2),N2) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z2),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z2),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_7506_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),N2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_7507_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z2),N2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z2),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% pochhammer_product
tff(fact_7508_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_7509_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_7510_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_7511_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_7512_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_7513_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N2))),N2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N2)),aa_nat_Code_integer(semiri3624122377584611663nteger,N2)) ).

% pochhammer_same
tff(fact_7514_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),N2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% pochhammer_same
tff(fact_7515_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))),N2) = 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)),N2)),aa_nat_int(semiri1406184849735516958ct_int,N2)) ).

% pochhammer_same
tff(fact_7516_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N2))),N2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)),aa_nat_real(semiri2265585572941072030t_real,N2)) ).

% pochhammer_same
tff(fact_7517_pochhammer__same,axiom,
    ! [N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N2))),N2) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)),aa_nat_complex(semiri5044797733671781792omplex,N2)) ).

% pochhammer_same
tff(fact_7518_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_7519_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_7520_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_7521_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_7522_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_7523_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_7524_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_7525_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_7526_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_7527_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_7528_fact__double,axiom,
    ! [N2: 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))),N2)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),N2))),aa_nat_rat(semiri773545260158071498ct_rat,N2)) ).

% fact_double
tff(fact_7529_fact__double,axiom,
    ! [N2: 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))),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),N2))),aa_nat_real(semiri2265585572941072030t_real,N2)) ).

% fact_double
tff(fact_7530_fact__double,axiom,
    ! [N2: 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))),N2)) = 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))),N2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),N2))),aa_nat_complex(semiri5044797733671781792omplex,N2)) ).

% fact_double
tff(fact_7531_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: complex,N2: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z2),aa_nat_nat(suc,N2))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,N2))) = groups6464643781859351333omplex(aTP_Lamm_lq(Z2),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))),N2)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7532_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: real,N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(comm_s7457072308508201937r_real(Z2),aa_nat_nat(suc,N2))),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,N2))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_lr(Z2)),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))),N2)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7533_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: rat,N2: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(suc,N2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,N2))) = groups73079841787564623at_rat(aTP_Lamm_ls(Z2),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))),N2)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7534_pochhammer__code,axiom,
    ! [N2: nat,A: complex] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N2) = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),N2) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_lt(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_complex) ) ) ) ).

% pochhammer_code
tff(fact_7535_pochhammer__code,axiom,
    ! [N2: nat,A: real] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N2) = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_real(comm_s7457072308508201937r_real(A),N2) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_lu(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_real) ) ) ) ).

% pochhammer_code
tff(fact_7536_pochhammer__code,axiom,
    ! [N2: nat,A: rat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_lv(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_rat) ) ) ) ).

% pochhammer_code
tff(fact_7537_pochhammer__code,axiom,
    ! [N2: nat,A: int] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N2) = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_int(comm_s4660882817536571857er_int(A),N2) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_lw(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_int) ) ) ) ).

% pochhammer_code
tff(fact_7538_pochhammer__code,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_lx(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),one_one_nat) ) ) ) ).

% pochhammer_code
tff(fact_7539_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_complex(semiri8010041392384452111omplex,N2) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_ly,N2),zero_zero_complex) ).

% of_nat_code
tff(fact_7540_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_real(semiri5074537144036343181t_real,N2) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_lz),N2),zero_zero_real) ).

% of_nat_code
tff(fact_7541_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_rat(semiri681578069525770553at_rat,N2) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_ma,N2),zero_zero_rat) ).

% of_nat_code
tff(fact_7542_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N2) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_mb),N2),zero_zero_nat) ).

% of_nat_code
tff(fact_7543_of__nat__code,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,N2) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_mc,N2),zero_zero_int) ).

% of_nat_code
tff(fact_7544_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_md(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7545_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_me(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7546_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mf(R2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_7547_central__binomial__lower__bound,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),N2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N2)))),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))),N2)),N2)))) ) ).

% central_binomial_lower_bound
tff(fact_7548_binomial__Suc__n,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),N2) = aa_nat_nat(suc,N2) ).

% binomial_Suc_n
tff(fact_7549_binomial__n__n,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),N2) = one_one_nat ).

% binomial_n_n
tff(fact_7550_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_int] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_set_int_bool(member_int(aa_complex_int(F,X4)),ring_1_Ints_int)) )
     => pp(aa_set_int_bool(member_int(groups858564598930262913ex_int(F,A2)),ring_1_Ints_int)) ) ).

% Ints_prod
tff(fact_7551_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_int] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_set_int_bool(member_int(aa_real_int(F,X4)),ring_1_Ints_int)) )
     => pp(aa_set_int_bool(member_int(groups4694064378042380927al_int(F,A2)),ring_1_Ints_int)) ) ).

% Ints_prod
tff(fact_7552_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => member_real(aa_complex_real(F,X4),ring_1_Ints_real) )
     => member_real(groups766887009212190081x_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_7553_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => member_real(aa_real_real(F,X4),ring_1_Ints_real) )
     => member_real(groups1681761925125756287l_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_7554_Ints__prod,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => member_real(aa_nat_real(F,X4),ring_1_Ints_real) )
     => member_real(aa_set_nat_real(groups129246275422532515t_real(F),A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_7555_Ints__prod,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => member_real(aa_int_real(F,X4),ring_1_Ints_real) )
     => member_real(groups2316167850115554303t_real(F,A2),ring_1_Ints_real) ) ).

% Ints_prod
tff(fact_7556_Ints__prod,axiom,
    ! [A2: set_complex,F: fun_complex_complex] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_set_complex_bool(member_complex(aa_complex_complex(F,X4)),ring_1_Ints_complex)) )
     => pp(aa_set_complex_bool(member_complex(groups3708469109370488835omplex(F,A2)),ring_1_Ints_complex)) ) ).

% Ints_prod
tff(fact_7557_Ints__prod,axiom,
    ! [A2: set_real,F: fun_real_complex] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_set_complex_bool(member_complex(aa_real_complex(F,X4)),ring_1_Ints_complex)) )
     => pp(aa_set_complex_bool(member_complex(groups713298508707869441omplex(F,A2)),ring_1_Ints_complex)) ) ).

% Ints_prod
tff(fact_7558_Ints__prod,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_set_complex_bool(member_complex(aa_nat_complex(F,X4)),ring_1_Ints_complex)) )
     => pp(aa_set_complex_bool(member_complex(groups6464643781859351333omplex(F,A2)),ring_1_Ints_complex)) ) ).

% Ints_prod
tff(fact_7559_Ints__prod,axiom,
    ! [A2: set_int,F: fun_int_complex] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_set_complex_bool(member_complex(aa_int_complex(F,X4)),ring_1_Ints_complex)) )
     => pp(aa_set_complex_bool(member_complex(groups7440179247065528705omplex(F,A2)),ring_1_Ints_complex)) ) ).

% Ints_prod
tff(fact_7560_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mg),A2) = one_one_nat ).

% prod.neutral_const
tff(fact_7561_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : groups705719431365010083at_int(aTP_Lamm_mh,A2) = one_one_int ).

% prod.neutral_const
tff(fact_7562_prod_Oneutral__const,axiom,
    ! [A2: set_int] : groups1705073143266064639nt_int(aTP_Lamm_mi,A2) = one_one_int ).

% prod.neutral_const
tff(fact_7563_of__nat__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_ey(F),A2) ).

% of_nat_prod
tff(fact_7564_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups6464643781859351333omplex(aTP_Lamm_ez(F),A2) ).

% of_nat_prod
tff(fact_7565_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_fd(F)),A2) ).

% of_nat_prod
tff(fact_7566_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups73079841787564623at_rat(aTP_Lamm_fa(F),A2) ).

% of_nat_prod
tff(fact_7567_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fc(F)),A2) ).

% of_nat_prod
tff(fact_7568_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_fb(F),A2) ).

% of_nat_prod
tff(fact_7569_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7570_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups2316167850115554303t_real(F,A2) = zero_zero_real )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7571_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups766887009212190081x_real(F,A2) = zero_zero_real )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7572_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups97031904164794029t_real(F,A2) = zero_zero_real )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
            & ( aa_Ext2541572767299617110t_real(F,X3) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7573_prod__zero__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( groups73079841787564623at_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
            & ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7574_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7575_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7576_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2245840878043517529at_rat(F,A2) = zero_zero_rat )
      <=> ? [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
            & ( aa_Extended_enat_rat(F,X3) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7577_prod__zero__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat )
      <=> ? [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
            & ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7578_prod__zero__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups861055069439313189ex_nat(F,A2) = zero_zero_nat )
      <=> ? [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
            & ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7579_prod_Oempty,axiom,
    ! [G: fun_Ex3741802288643809319omplex] : groups4622424608036095791omplex(G,bot_bo7653980558646680370d_enat) = one_one_complex ).

% prod.empty
tff(fact_7580_prod_Oempty,axiom,
    ! [G: fun_Ex1451620480498224165t_real] : groups97031904164794029t_real(G,bot_bo7653980558646680370d_enat) = one_one_real ).

% prod.empty
tff(fact_7581_prod_Oempty,axiom,
    ! [G: fun_Ex4152069839811072081at_rat] : groups2245840878043517529at_rat(G,bot_bo7653980558646680370d_enat) = one_one_rat ).

% prod.empty
tff(fact_7582_prod_Oempty,axiom,
    ! [G: fun_Ex7660298398204468809at_nat] : groups2880970938130013265at_nat(G,bot_bo7653980558646680370d_enat) = one_one_nat ).

% prod.empty
tff(fact_7583_prod_Oempty,axiom,
    ! [G: fun_Ex3482447378695272101at_int] : groups2878480467620962989at_int(G,bot_bo7653980558646680370d_enat) = one_one_int ).

% prod.empty
tff(fact_7584_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_7585_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_7586_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_7587_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_7588_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_7589_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7590_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7591_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7592_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4622424608036095791omplex(G,A2) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7593_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_real] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_7594_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_7595_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_7596_prod_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups97031904164794029t_real(G,A2) = one_one_real ) ) ).

% prod.infinite
tff(fact_7597_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups73079841787564623at_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7598_prod_Oinfinite,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( ~ pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7599_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(gbinomial_real(zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_7600_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_7601_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_7602_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_7603_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_7604_binomial__1,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,zero_zero_nat)) = N2 ).

% binomial_1
tff(fact_7605_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K)) ) ).

% binomial_eq_0_iff
tff(fact_7606_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_7607_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(gbinomial_real(A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_7608_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_7609_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_7610_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_7611_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_7612_binomial__n__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_7613_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_mj(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_mj(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7614_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_mk(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_mk(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7615_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ml(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ml(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7616_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_mm(A),B),S3) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_mm(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7617_prod_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4622424608036095791omplex(aa_fun616941850244831356omplex(aTP_Lamm_mn(A),B),S3) = aa_Ext8748583412541189208omplex(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4622424608036095791omplex(aa_fun616941850244831356omplex(aTP_Lamm_mn(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta
tff(fact_7618_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7619_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mp(A),B)),S3) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mp(A),B)),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7620_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mq(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mq(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7621_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_mr(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_mr(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7622_prod_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups97031904164794029t_real(aa_fun1749635019461158908t_real(aTP_Lamm_ms(A),B),S3) = aa_Ext2541572767299617110t_real(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups97031904164794029t_real(aa_fun1749635019461158908t_real(aTP_Lamm_ms(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta
tff(fact_7623_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_mt(A),B),S3) = aa_real_complex(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_mt(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7624_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_mu(A),B),S3) = aa_nat_complex(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_mu(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7625_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_mv(A),B),S3) = aa_int_complex(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_mv(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7626_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_mw(A),B),S3) = aa_complex_complex(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_mw(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7627_prod_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4622424608036095791omplex(aa_fun616941850244831356omplex(aTP_Lamm_mx(A),B),S3) = aa_Ext8748583412541189208omplex(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups4622424608036095791omplex(aa_fun616941850244831356omplex(aTP_Lamm_mx(A),B),S3) = one_one_complex ) ) ) ) ).

% prod.delta'
tff(fact_7628_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( ( member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(A),B),S3) = aa_real_real(B,A) ) )
        & ( ~ member_real(A,S3)
         => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7629_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( ( pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mz(A),B)),S3) = aa_nat_real(B,A) ) )
        & ( ~ pp(aa_set_nat_bool(member_nat(A),S3))
         => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mz(A),B)),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7630_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,S3))
     => ( ( pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_na(A),B),S3) = aa_int_real(B,A) ) )
        & ( ~ pp(aa_set_int_bool(member_int(A),S3))
         => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_na(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7631_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
     => ( ( pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nb(A),B),S3) = aa_complex_real(B,A) ) )
        & ( ~ pp(aa_set_complex_bool(member_complex(A),S3))
         => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nb(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7632_prod_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
     => ( ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups97031904164794029t_real(aa_fun1749635019461158908t_real(aTP_Lamm_nc(A),B),S3) = aa_Ext2541572767299617110t_real(B,A) ) )
        & ( ~ pp(aa_set3347977610044113549t_bool(member_Extended_enat(A),S3))
         => ( groups97031904164794029t_real(aa_fun1749635019461158908t_real(aTP_Lamm_nc(A),B),S3) = one_one_real ) ) ) ) ).

% prod.delta'
tff(fact_7633_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) ) ).

% zero_less_binomial_iff
tff(fact_7634_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_real(G,N2)) ).

% prod.lessThan_Suc
tff(fact_7635_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_rat(G,N2)) ).

% prod.lessThan_Suc
tff(fact_7636_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_nat(G,N2)) ).

% prod.lessThan_Suc
tff(fact_7637_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N2))),aa_nat_int(G,N2)) ).

% prod.lessThan_Suc
tff(fact_7638_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_complex] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = one_one_complex ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_complex(G,aa_nat_nat(suc,N2))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7639_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_real] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = one_one_real ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7640_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_rat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = one_one_rat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7641_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = one_one_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7642_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: fun_nat_int] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = one_one_int ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),M))
       => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_7643_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_complex,A2: set_complex] :
      ( ( groups3708469109370488835omplex(G,A2) != one_one_complex )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A2))
           => ( aa_complex_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7644_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A2: set_real] :
      ( ( groups713298508707869441omplex(G,A2) != one_one_complex )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7645_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A2: set_nat] :
      ( ( groups6464643781859351333omplex(G,A2) != one_one_complex )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7646_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A2: set_int] :
      ( ( groups7440179247065528705omplex(G,A2) != one_one_complex )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_complex(G,A3) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7647_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A2: set_complex] :
      ( ( groups766887009212190081x_real(G,A2) != one_one_real )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A2))
           => ( aa_complex_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7648_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A2: set_real] :
      ( ( groups1681761925125756287l_real(G,A2) != one_one_real )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7649_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A2: set_nat] :
      ( ( aa_set_nat_real(groups129246275422532515t_real(G),A2) != one_one_real )
     => ~ ! [A3: nat] :
            ( pp(aa_set_nat_bool(member_nat(A3),A2))
           => ( aa_nat_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7650_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A2: set_int] :
      ( ( groups2316167850115554303t_real(G,A2) != one_one_real )
     => ~ ! [A3: int] :
            ( pp(aa_set_int_bool(member_int(A3),A2))
           => ( aa_int_real(G,A3) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7651_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A2: set_complex] :
      ( ( groups225925009352817453ex_rat(G,A2) != one_one_rat )
     => ~ ! [A3: complex] :
            ( pp(aa_set_complex_bool(member_complex(A3),A2))
           => ( aa_complex_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7652_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A2: set_real] :
      ( ( groups4061424788464935467al_rat(G,A2) != one_one_rat )
     => ~ ! [A3: real] :
            ( member_real(A3,A2)
           => ( aa_real_rat(G,A3) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_7653_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_nat(G,X4) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A2) = one_one_nat ) ) ).

% prod.neutral
tff(fact_7654_prod_Oneutral,axiom,
    ! [A2: set_nat,G: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( aa_nat_int(G,X4) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_7655_prod_Oneutral,axiom,
    ! [A2: set_int,G: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( aa_int_int(G,X4) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).

% prod.neutral
tff(fact_7656_choose__one,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),one_one_nat) = N2 ).

% choose_one
tff(fact_7657_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I2)),aa_complex_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2))) ) ).

% prod_mono
tff(fact_7658_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2))) ) ).

% prod_mono
tff(fact_7659_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),aa_nat_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2))) ) ).

% prod_mono
tff(fact_7660_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2))) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2))) ) ).

% prod_mono
tff(fact_7661_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(G,A2))) ) ).

% prod_mono
tff(fact_7662_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(G,A2))) ) ).

% prod_mono
tff(fact_7663_prod__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( pp(aa_set_nat_bool(member_nat(I2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),groups73079841787564623at_rat(G,A2))) ) ).

% prod_mono
tff(fact_7664_prod__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( pp(aa_set_int_bool(member_int(I2),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(G,A2))) ) ).

% prod_mono
tff(fact_7665_prod__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( pp(aa_set_complex_bool(member_complex(I2),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups861055069439313189ex_nat(F,A2)),groups861055069439313189ex_nat(G,A2))) ) ).

% prod_mono
tff(fact_7666_prod__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),groups4696554848551431203al_nat(G,A2))) ) ).

% prod_mono
tff(fact_7667_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_nonneg
tff(fact_7668_prod__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A2))) ) ).

% prod_nonneg
tff(fact_7669_prod__nonneg,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_nonneg
tff(fact_7670_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2))) ) ).

% prod_pos
tff(fact_7671_prod__pos,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A2))) ) ).

% prod_pos
tff(fact_7672_prod__pos,axiom,
    ! [A2: set_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(F,X4))) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A2))) ) ).

% prod_pos
tff(fact_7673_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A2))) ) ).

% prod_ge_1
tff(fact_7674_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A2))) ) ).

% prod_ge_1
tff(fact_7675_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),A2))) ) ).

% prod_ge_1
tff(fact_7676_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,X4))) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A2))) ) ).

% prod_ge_1
tff(fact_7677_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_7678_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_7679_prod__ge__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_7680_prod__ge__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X4))) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A2))) ) ).

% prod_ge_1
tff(fact_7681_prod__ge__1,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_complex_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups861055069439313189ex_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_7682_prod__ge__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X4))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A2))) ) ).

% prod_ge_1
tff(fact_7683_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & ( aa_nat_real(F,X2) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7684_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_real(F,X2) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7685_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_real(F,X2) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7686_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ? [X2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
            & ( aa_Ext2541572767299617110t_real(F,X2) = zero_zero_real ) )
       => ( groups97031904164794029t_real(F,A2) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_7687_prod__zero,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ? [X2: nat] :
            ( pp(aa_set_nat_bool(member_nat(X2),A2))
            & ( aa_nat_rat(F,X2) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7688_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_rat(F,X2) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7689_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_rat(F,X2) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7690_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ? [X2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X2),A2))
            & ( aa_Extended_enat_rat(F,X2) = zero_zero_rat ) )
       => ( groups2245840878043517529at_rat(F,A2) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_7691_prod__zero,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ? [X2: int] :
            ( pp(aa_set_int_bool(member_int(X2),A2))
            & ( aa_int_nat(F,X2) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_7692_prod__zero,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ? [X2: complex] :
            ( pp(aa_set_complex_bool(member_complex(X2),A2))
            & ( aa_complex_nat(F,X2) = zero_zero_nat ) )
       => ( groups861055069439313189ex_nat(F,A2) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_7693_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_7694_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_nd(F),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_7695_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups129246275422532515t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ne(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_7696_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_nf(F),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_7697_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ng(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_7698_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_nh(F),A,B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_7699_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_7700_Suc__times__binomial,axiom,
    ! [K: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) ).

% Suc_times_binomial
tff(fact_7701_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% binomial_symmetric
tff(fact_7702_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R2)),M)) ).

% choose_mult_lemma
tff(fact_7703_binomial__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N2),R2))) ) ).

% binomial_le_pow
tff(fact_7704_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups713298508707869441omplex(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_gf(A2),P))) = groups713298508707869441omplex(aa_fun6813637976755515871omplex(aTP_Lamm_ni(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7705_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( groups6464643781859351333omplex(G,collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_go(A2),P))) = groups6464643781859351333omplex(aa_fun8588439219086927967omplex(aTP_Lamm_nj(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7706_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_gh(A2),P))) = groups7440179247065528705omplex(aa_fun3606824691168040799omplex(aTP_Lamm_nk(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7707_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_gj(A2),P))) = groups3708469109370488835omplex(aa_fun1034237174562290143omplex(aTP_Lamm_nl(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7708_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3741802288643809319omplex,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4622424608036095791omplex(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_gl(A2),P))) = groups4622424608036095791omplex(aa_fun1711458463208245493omplex(aTP_Lamm_nm(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7709_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_bool] :
      ( finite_finite_real(A2)
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun6476784834714468668l_bool(aTP_Lamm_gf(A2),P))) = groups1681761925125756287l_real(aa_fun6348708079608492509l_real(aTP_Lamm_nn(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7710_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_real,P: fun_nat_bool] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),collect_nat(aa_fun3878383735219268284t_bool(aTP_Lamm_go(A2),P))) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1750223370215434333t_real(aTP_Lamm_no(G),P)),A2) ) ) ).

% prod.inter_filter
tff(fact_7711_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_bool] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun5146555767781538748t_bool(aTP_Lamm_gh(A2),P))) = groups2316167850115554303t_real(aa_fun6974987029372589917t_real(aTP_Lamm_np(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7712_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun1096865461735438652x_bool(aTP_Lamm_gj(A2),P))) = groups766887009212190081x_real(aa_fun2902006202505214429x_real(aTP_Lamm_nq(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7713_prod_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,P: fun_Ex8563670519792538868t_bool] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups97031904164794029t_real(G,collec4429806609662206161d_enat(aa_fun4759809662886658982t_bool(aTP_Lamm_gl(A2),P))) = groups97031904164794029t_real(aa_fun1090353380788292083t_real(aTP_Lamm_nr(G),P),A2) ) ) ).

% prod.inter_filter
tff(fact_7714_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7715_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N2))) = groups705719431365010083at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_7716_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hl(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7717_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ns(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_7718_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7719_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7720_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7721_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X4)))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X4)),one_one_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),one_one_real)) ) ).

% prod_le_1
tff(fact_7722_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7723_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7724_prod__le__1,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups73079841787564623at_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7725_prod__le__1,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X4: int] :
          ( pp(aa_set_int_bool(member_int(X4),A2))
         => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X4)))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X4)),one_one_rat)) ) )
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),one_one_rat)) ) ).

% prod_le_1
tff(fact_7726_prod__le__1,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X4: complex] :
          ( pp(aa_set_complex_bool(member_complex(X4),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups861055069439313189ex_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7727_prod__le__1,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X4)))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X4)),one_one_nat)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4696554848551431203al_nat(F,A2)),one_one_nat)) ) ).

% prod_le_1
tff(fact_7728_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_nat_complex(H,X4)),aa_nat_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups6464643781859351333omplex(H,S3)),groups6464643781859351333omplex(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7729_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_int_complex(H,X4)),aa_int_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups7440179247065528705omplex(H,S3)),groups7440179247065528705omplex(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7730_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(H,X4)),aa_complex_complex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups3708469109370488835omplex(H,S3)),groups3708469109370488835omplex(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7731_prod_Orelated,axiom,
    ! [R: fun_co1378794544173401837x_bool,S3: set_Extended_enat,H: fun_Ex3741802288643809319omplex,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,one_one_complex),one_one_complex))
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,X1),X23))
              & pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,Y1),Y23)) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,aa_Ext8748583412541189208omplex(H,X4)),aa_Ext8748583412541189208omplex(G,X4))) )
           => pp(aa_complex_bool(aa_com1553036987509291606x_bool(R,groups4622424608036095791omplex(H,S3)),groups4622424608036095791omplex(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7732_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S3: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_nat_real(H,X4)),aa_nat_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_set_nat_real(groups129246275422532515t_real(H),S3)),aa_set_nat_real(groups129246275422532515t_real(G),S3))) ) ) ) ) ).

% prod.related
tff(fact_7733_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_int_real(H,X4)),aa_int_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups2316167850115554303t_real(H,S3)),groups2316167850115554303t_real(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7734_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S3))
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_complex_real(H,X4)),aa_complex_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups766887009212190081x_real(H,S3)),groups766887009212190081x_real(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7735_prod_Orelated,axiom,
    ! [R: fun_re1255401033344146673l_bool,S3: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,one_one_real),one_one_real))
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,X1),X23))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,Y1),Y23)) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23))) )
       => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,S3))
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,aa_Ext2541572767299617110t_real(H,X4)),aa_Ext2541572767299617110t_real(G,X4))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(R,groups97031904164794029t_real(H,S3)),groups97031904164794029t_real(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7736_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23))) )
       => ( pp(aa_set_nat_bool(finite_finite_nat,S3))
         => ( ! [X4: nat] :
                ( pp(aa_set_nat_bool(member_nat(X4),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_nat_rat(H,X4)),aa_nat_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups73079841787564623at_rat(H,S3)),groups73079841787564623at_rat(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7737_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_bool,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,one_one_rat),one_one_rat))
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R,X1),X23))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(R,Y1),Y23)) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23))) )
       => ( pp(aa_set_int_bool(finite_finite_int,S3))
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,aa_int_rat(H,X4)),aa_int_rat(G,X4))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(R,groups1072433553688619179nt_rat(H,S3)),groups1072433553688619179nt_rat(G,S3))) ) ) ) ) ).

% prod.related
tff(fact_7738_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [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(T4,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T4,T3))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T4,T3))
                   => 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,T3)
                       => ( 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,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7739_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T4: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( finite_finite_real(S4)
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [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))
               => pp(aa_set_int_bool(member_int(aa_real_int(J,A3)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => 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] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( 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,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7740_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S3: set_real,I: fun_complex_real,J: fun_real_complex,T4: set_complex,G: fun_real_complex,H: fun_complex_complex] :
      ( finite_finite_real(S4)
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [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))
               => pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => 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] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( 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,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7741_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_Extended_enat,S3: set_real,I: fun_Ex1451620480498224165t_real,J: fun_re6065469302365115155d_enat,T4: set_Extended_enat,G: fun_real_complex,H: fun_Ex3741802288643809319omplex] :
      ( finite_finite_real(S4)
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_Ext2541572767299617110t_real(I,aa_rea1194391308913012860d_enat(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_rea1194391308913012860d_enat(J,A3)),minus_925952699566721837d_enat(T4,T3))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                 => ( aa_rea1194391308913012860d_enat(J,aa_Ext2541572767299617110t_real(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                   => member_real(aa_Ext2541572767299617110t_real(I,B3),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T3))
                       => ( aa_Ext8748583412541189208omplex(H,B3) = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_Ext8748583412541189208omplex(H,aa_rea1194391308913012860d_enat(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups713298508707869441omplex(G,S3) = groups4622424608036095791omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7742_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S3: set_int,I: fun_real_int,J: fun_int_real,T4: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( finite_finite_real(T3)
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_real_int(I,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T4,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T4,T3))
                 => ( aa_int_real(J,aa_real_int(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T4,T3))
                   => pp(aa_set_int_bool(member_int(aa_real_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups713298508707869441omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7743_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S3: set_int,I: fun_int_int,J: fun_int_int,T4: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_int_int(I,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => pp(aa_set_int_bool(member_int(aa_int_int(J,A3)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_int_int(J,aa_int_int(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_int_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7744_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_complex,S3: set_int,I: fun_complex_int,J: fun_int_complex,T4: set_complex,G: fun_int_complex,H: fun_complex_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_complex_int(I,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A3)),minus_811609699411566653omplex(T4,T3))) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                 => ( aa_int_complex(J,aa_complex_int(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_complex_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( pp(aa_set_complex_bool(member_complex(B3),T3))
                       => ( aa_complex_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_complex_complex(H,aa_int_complex(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups3708469109370488835omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7745_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_Extended_enat,S3: set_int,I: fun_Ex3482447378695272101at_int,J: fun_in5914218760682845459d_enat,T4: set_Extended_enat,G: fun_int_complex,H: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set_int_bool(finite_finite_int,S4))
     => ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T3))
       => ( ! [A3: int] :
              ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
             => ( aa_Extended_enat_int(I,aa_int_Extended_enat(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(S3,S4)))
               => pp(aa_set3347977610044113549t_bool(member_Extended_enat(aa_int_Extended_enat(J,A3)),minus_925952699566721837d_enat(T4,T3))) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                 => ( aa_int_Extended_enat(J,aa_Extended_enat_int(I,B3)) = B3 ) )
             => ( ! [B3: extended_enat] :
                    ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(T4,T3)))
                   => pp(aa_set_int_bool(member_int(aa_Extended_enat_int(I,B3)),minus_minus_set_int(S3,S4))) )
               => ( ! [A3: int] :
                      ( pp(aa_set_int_bool(member_int(A3),S4))
                     => ( aa_int_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: extended_enat] :
                        ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),T3))
                       => ( aa_Ext8748583412541189208omplex(H,B3) = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( pp(aa_set_int_bool(member_int(A3),S3))
                         => ( aa_Ext8748583412541189208omplex(H,aa_int_Extended_enat(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups7440179247065528705omplex(G,S3) = groups4622424608036095791omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7746_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_real,S3: set_complex,I: fun_real_complex,J: fun_complex_real,T4: set_real,G: fun_complex_complex,H: fun_real_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
     => ( finite_finite_real(T3)
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
             => ( aa_real_complex(I,aa_complex_real(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
               => member_real(aa_complex_real(J,A3),minus_minus_set_real(T4,T3)) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(T4,T3))
                 => ( aa_complex_real(J,aa_real_complex(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( member_real(B3,minus_minus_set_real(T4,T3))
                   => pp(aa_set_complex_bool(member_complex(aa_real_complex(I,B3)),minus_811609699411566653omplex(S3,S4))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S4))
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( member_real(B3,T3)
                       => ( aa_real_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S3))
                         => ( aa_real_complex(H,aa_complex_real(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S3) = groups713298508707869441omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7747_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_int,S3: set_complex,I: fun_int_complex,J: fun_complex_int,T4: set_int,G: fun_complex_complex,H: fun_int_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S4))
     => ( pp(aa_set_int_bool(finite_finite_int,T3))
       => ( ! [A3: complex] :
              ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
             => ( aa_int_complex(I,aa_complex_int(J,A3)) = A3 ) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(S3,S4)))
               => pp(aa_set_int_bool(member_int(aa_complex_int(J,A3)),minus_minus_set_int(T4,T3))) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                 => ( aa_complex_int(J,aa_int_complex(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(T4,T3)))
                   => pp(aa_set_complex_bool(member_complex(aa_int_complex(I,B3)),minus_811609699411566653omplex(S3,S4))) )
               => ( ! [A3: complex] :
                      ( pp(aa_set_complex_bool(member_complex(A3),S4))
                     => ( aa_complex_complex(G,A3) = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( pp(aa_set_int_bool(member_int(B3),T3))
                       => ( aa_int_complex(H,B3) = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( pp(aa_set_complex_bool(member_complex(A3),S3))
                         => ( aa_int_complex(H,aa_complex_int(J,A3)) = aa_complex_complex(G,A3) ) )
                     => ( groups3708469109370488835omplex(G,S3) = groups7440179247065528705omplex(H,T4) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_7748_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))) ) ).

% zero_less_binomial
tff(fact_7749_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_7750_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_7751_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_7752_binomial__absorb__comp,axiom,
    ! [N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_7753_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7754_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7755_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7756_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),K) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7757_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N2)),K) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7758_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7759_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_complex] :
      ( finite_finite_real(A2)
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_nt(G)))) = groups713298508707869441omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7760_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups7440179247065528705omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_nu(G)))) = groups7440179247065528705omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7761_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_nv(G)))) = groups3708469109370488835omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7762_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups4622424608036095791omplex(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_nw(G)))) = groups4622424608036095791omplex(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7763_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( groups1681761925125756287l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_nx(G)))) = groups1681761925125756287l_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7764_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ny(G)))) = groups2316167850115554303t_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7765_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_nz(G)))) = groups766887009212190081x_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7766_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( groups97031904164794029t_real(G,minus_925952699566721837d_enat(A2,collec4429806609662206161d_enat(aTP_Lamm_oa(G)))) = groups97031904164794029t_real(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7767_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( groups4061424788464935467al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_ob(G)))) = groups4061424788464935467al_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7768_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_oc(G)))) = groups1072433553688619179nt_rat(G,A2) ) ) ).

% prod.setdiff_irrelevant
tff(fact_7769_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ki(G),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nat_diff_reindex
tff(fact_7770_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_od(G),N2),aa_nat_set_nat(set_ord_lessThan_nat,N2)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nat_diff_reindex
tff(fact_7771_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N2: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hu(G),N2),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) ).

% prod.atLeastAtMost_rev
tff(fact_7772_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,N2: nat,M: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_oe(G),N2),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M)) ).

% prod.atLeastAtMost_rev
tff(fact_7773_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_of(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7774_gbinomial__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_og(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7775_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(aTP_Lamm_oh(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7776_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oi,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7777_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K)) = aa_int_int(divide_divide_int(groups705719431365010083at_int(aTP_Lamm_oj(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_7778_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I)))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7779_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I)))
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7780_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7781_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7782_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,I)))
         => ( ! [I2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,I2))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups97031904164794029t_real(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7783_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I)))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7784_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( pp(aa_set_nat_bool(member_nat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I)))
         => ( ! [I2: nat] :
                ( pp(aa_set_nat_bool(member_nat(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7785_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( pp(aa_set_int_bool(member_int(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I)))
         => ( ! [I2: int] :
                ( pp(aa_set_int_bool(member_int(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7786_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( pp(aa_set_complex_bool(member_complex(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I)))
         => ( ! [I2: complex] :
                ( pp(aa_set_complex_bool(member_complex(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7787_less__1__prod2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I),I5))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_Extended_enat_rat(F,I)))
         => ( ! [I2: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_Extended_enat_rat(F,I2))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups2245840878043517529at_rat(F,I5))) ) ) ) ) ).

% less_1_prod2
tff(fact_7788_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_complex_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7789_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_Ext2541572767299617110t_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups97031904164794029t_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7790_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7791_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5))) ) ) ) ).

% less_1_prod
tff(fact_7792_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(F,I2))) )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7793_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( pp(aa_set_complex_bool(member_complex(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_complex_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7794_less__1__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_Extended_enat_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups2245840878043517529at_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7795_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_real_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7796_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( pp(aa_set_nat_bool(member_nat(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7797_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( pp(aa_set_int_bool(member_int(I2),I5))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(F,I2))) )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5))) ) ) ) ).

% less_1_prod
tff(fact_7798_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A2) = groups713298508707869441omplex(H,B2) )
              <=> ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7799_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_complex(H,B3) = one_one_complex ) )
             => ( ( groups7440179247065528705omplex(G,A2) = groups7440179247065528705omplex(H,B2) )
              <=> ( groups7440179247065528705omplex(G,C4) = groups7440179247065528705omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7800_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A2) = groups3708469109370488835omplex(H,B2) )
              <=> ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7801_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex3741802288643809319omplex,H: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext8748583412541189208omplex(G,A3) = one_one_complex ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Ext8748583412541189208omplex(H,B3) = one_one_complex ) )
             => ( ( groups4622424608036095791omplex(G,A2) = groups4622424608036095791omplex(H,B2) )
              <=> ( groups4622424608036095791omplex(G,C4) = groups4622424608036095791omplex(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7802_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B2) )
              <=> ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7803_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_real(G,A3) = one_one_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_real(H,B3) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,A2) = groups2316167850115554303t_real(H,B2) )
              <=> ( groups2316167850115554303t_real(G,C4) = groups2316167850115554303t_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7804_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A2) = groups766887009212190081x_real(H,B2) )
              <=> ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7805_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext2541572767299617110t_real(G,A3) = one_one_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = one_one_real ) )
             => ( ( groups97031904164794029t_real(G,A2) = groups97031904164794029t_real(H,B2) )
              <=> ( groups97031904164794029t_real(G,C4) = groups97031904164794029t_real(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7806_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A2) = groups4061424788464935467al_rat(H,B2) )
              <=> ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7807_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_rat(H,B3) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,A2) = groups1072433553688619179nt_rat(H,B2) )
              <=> ( groups1072433553688619179nt_rat(G,C4) = groups1072433553688619179nt_rat(H,C4) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_7808_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_complex(H,B3) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C4) = groups713298508707869441omplex(H,C4) )
               => ( groups713298508707869441omplex(G,A2) = groups713298508707869441omplex(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7809_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_complex(H,B3) = one_one_complex ) )
             => ( ( groups7440179247065528705omplex(G,C4) = groups7440179247065528705omplex(H,C4) )
               => ( groups7440179247065528705omplex(G,A2) = groups7440179247065528705omplex(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7810_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_complex(G,A3) = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_complex(H,B3) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C4) = groups3708469109370488835omplex(H,C4) )
               => ( groups3708469109370488835omplex(G,A2) = groups3708469109370488835omplex(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7811_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex3741802288643809319omplex,H: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext8748583412541189208omplex(G,A3) = one_one_complex ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Ext8748583412541189208omplex(H,B3) = one_one_complex ) )
             => ( ( groups4622424608036095791omplex(G,C4) = groups4622424608036095791omplex(H,C4) )
               => ( groups4622424608036095791omplex(G,A2) = groups4622424608036095791omplex(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7812_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_real(G,A3) = one_one_real ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_real(H,B3) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C4) = groups1681761925125756287l_real(H,C4) )
               => ( groups1681761925125756287l_real(G,A2) = groups1681761925125756287l_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7813_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_real(G,A3) = one_one_real ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_real(H,B3) = one_one_real ) )
             => ( ( groups2316167850115554303t_real(G,C4) = groups2316167850115554303t_real(H,C4) )
               => ( groups2316167850115554303t_real(G,A2) = groups2316167850115554303t_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7814_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
       => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),minus_811609699411566653omplex(C4,A2)))
               => ( aa_complex_real(G,A3) = one_one_real ) )
           => ( ! [B3: complex] :
                  ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(C4,B2)))
                 => ( aa_complex_real(H,B3) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C4) = groups766887009212190081x_real(H,C4) )
               => ( groups766887009212190081x_real(G,A2) = groups766887009212190081x_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7815_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,C4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),C4))
       => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(B2),C4))
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),minus_925952699566721837d_enat(C4,A2)))
               => ( aa_Ext2541572767299617110t_real(G,A3) = one_one_real ) )
           => ( ! [B3: extended_enat] :
                  ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(C4,B2)))
                 => ( aa_Ext2541572767299617110t_real(H,B3) = one_one_real ) )
             => ( ( groups97031904164794029t_real(G,C4) = groups97031904164794029t_real(H,C4) )
               => ( groups97031904164794029t_real(G,A2) = groups97031904164794029t_real(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7816_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C4)
     => ( ord_less_eq_set_real(A2,C4)
       => ( ord_less_eq_set_real(B2,C4)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C4,A2))
               => ( aa_real_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: real] :
                  ( member_real(B3,minus_minus_set_real(C4,B2))
                 => ( aa_real_rat(H,B3) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C4) = groups4061424788464935467al_rat(H,C4) )
               => ( groups4061424788464935467al_rat(G,A2) = groups4061424788464935467al_rat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7817_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,C4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
       => ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),minus_minus_set_int(C4,A2)))
               => ( aa_int_rat(G,A3) = one_one_rat ) )
           => ( ! [B3: int] :
                  ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(C4,B2)))
                 => ( aa_int_rat(H,B3) = one_one_rat ) )
             => ( ( groups1072433553688619179nt_rat(G,C4) = groups1072433553688619179nt_rat(H,C4) )
               => ( groups1072433553688619179nt_rat(G,A2) = groups1072433553688619179nt_rat(H,B2) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_7818_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7819_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7820_prod_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = one_one_complex ) )
         => ( groups4622424608036095791omplex(G,S3) = groups4622424608036095791omplex(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7821_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7822_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7823_prod_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = one_one_real ) )
         => ( groups97031904164794029t_real(G,S3) = groups97031904164794029t_real(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7824_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G,X4) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,S3) = groups1072433553688619179nt_rat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7825_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S3) = groups225925009352817453ex_rat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7826_prod_Omono__neutral__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_rat(G,X4) = one_one_rat ) )
         => ( groups2245840878043517529at_rat(G,S3) = groups2245840878043517529at_rat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7827_prod_Omono__neutral__left,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,S3) = groups1707563613775114915nt_nat(G,T4) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_7828_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,T4) = groups7440179247065528705omplex(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7829_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T4) = groups3708469109370488835omplex(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7830_prod_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = one_one_complex ) )
         => ( groups4622424608036095791omplex(G,T4) = groups4622424608036095791omplex(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7831_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,T4) = groups2316167850115554303t_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7832_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T4) = groups766887009212190081x_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7833_prod_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = one_one_real ) )
         => ( groups97031904164794029t_real(G,T4) = groups97031904164794029t_real(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7834_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G,X4) = one_one_rat ) )
         => ( groups1072433553688619179nt_rat(G,T4) = groups1072433553688619179nt_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7835_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_rat(G,X4) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T4) = groups225925009352817453ex_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7836_prod_Omono__neutral__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Extended_enat_rat(G,X4) = one_one_rat ) )
         => ( groups2245840878043517529at_rat(G,T4) = groups2245840878043517529at_rat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7837_prod_Omono__neutral__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_nat(G,X4) = one_one_nat ) )
         => ( groups1707563613775114915nt_nat(G,T4) = groups1707563613775114915nt_nat(G,S3) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_7838_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7839_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
           => ( groups7440179247065528705omplex(G,S3) = groups7440179247065528705omplex(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7840_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(H,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,S3) = groups3708469109370488835omplex(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7841_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex3741802288643809319omplex,G: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext8748583412541189208omplex(H,X4) = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Ext8748583412541189208omplex(G,X4) = aa_Ext8748583412541189208omplex(H,X4) ) )
           => ( groups4622424608036095791omplex(G,S3) = groups4622424608036095791omplex(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7842_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7843_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(H,X4) = one_one_real ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups2316167850115554303t_real(G,S3) = groups2316167850115554303t_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7844_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(H,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,S3) = groups766887009212190081x_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7845_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,H: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(H,X4) = one_one_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups97031904164794029t_real(G,S3) = groups97031904164794029t_real(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7846_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7847_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_int,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(H,X4) = one_one_rat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups1072433553688619179nt_rat(G,S3) = groups1072433553688619179nt_rat(H,T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_7848_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) = groups713298508707869441omplex(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7849_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_complex(G,X4) = aa_int_complex(H,X4) ) )
           => ( groups7440179247065528705omplex(G,T4) = groups7440179247065528705omplex(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7850_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_complex(G,X4) = one_one_complex ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_complex(G,X4) = aa_complex_complex(H,X4) ) )
           => ( groups3708469109370488835omplex(G,T4) = groups3708469109370488835omplex(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7851_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex3741802288643809319omplex,H: fun_Ex3741802288643809319omplex] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext8748583412541189208omplex(G,X4) = one_one_complex ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Ext8748583412541189208omplex(G,X4) = aa_Ext8748583412541189208omplex(H,X4) ) )
           => ( groups4622424608036095791omplex(G,T4) = groups4622424608036095791omplex(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7852_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) = groups1681761925125756287l_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7853_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_real(G,X4) = one_one_real ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_real(G,X4) = aa_int_real(H,X4) ) )
           => ( groups2316167850115554303t_real(G,T4) = groups2316167850115554303t_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7854_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T4))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S3),T4))
       => ( ! [X4: complex] :
              ( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(T4,S3)))
             => ( aa_complex_real(G,X4) = one_one_real ) )
         => ( ! [X4: complex] :
                ( pp(aa_set_complex_bool(member_complex(X4),S3))
               => ( aa_complex_real(G,X4) = aa_complex_real(H,X4) ) )
           => ( groups766887009212190081x_real(G,T4) = groups766887009212190081x_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7855_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_Extended_enat,S3: set_Extended_enat,G: fun_Ex1451620480498224165t_real,H: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,T4))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(S3),T4))
       => ( ! [X4: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),minus_925952699566721837d_enat(T4,S3)))
             => ( aa_Ext2541572767299617110t_real(G,X4) = one_one_real ) )
         => ( ! [X4: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X4),S3))
               => ( aa_Ext2541572767299617110t_real(G,X4) = aa_Ext2541572767299617110t_real(H,X4) ) )
           => ( groups97031904164794029t_real(G,T4) = groups97031904164794029t_real(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7856_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T4)
     => ( ord_less_eq_set_real(S3,T4)
       => ( ! [X4: real] :
              ( member_real(X4,minus_minus_set_real(T4,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,T4) = groups4061424788464935467al_rat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7857_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_int,S3: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,T4))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(S3),T4))
       => ( ! [X4: int] :
              ( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(T4,S3)))
             => ( aa_int_rat(G,X4) = one_one_rat ) )
         => ( ! [X4: int] :
                ( pp(aa_set_int_bool(member_int(X4),S3))
               => ( aa_int_rat(G,X4) = aa_int_rat(H,X4) ) )
           => ( groups1072433553688619179nt_rat(G,T4) = groups1072433553688619179nt_rat(H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_7858_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7859_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7860_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7861_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ).

% prod.atLeast0_atMost_Suc
tff(fact_7862_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7863_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,N2))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7864_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,N2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7865_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,N2))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_7866_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7867_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7868_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7869_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_7870_binomial__absorption,axiom,
    ! [K: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_7871_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_7872_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_7873_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_7874_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorb_comp
tff(fact_7875_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_7876_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_7877_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(gbinomial_real(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7878_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K))) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7879_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_7880_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_7881_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_7882_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_7883_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(gbinomial_real(A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_7884_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_7885_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N2) ) ) ).

% binomial_fact_lemma
tff(fact_7886_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_7887_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_7888_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_7889_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.lessThan_Suc_shift
tff(fact_7890_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7891_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7892_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7893_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_7894_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.atLeast1_atMost_eq
tff(fact_7895_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2)) = groups705719431365010083at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.atLeast1_atMost_eq
tff(fact_7896_fact__prod,axiom,
    ! [N2: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N2) = aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2))) ).

% fact_prod
tff(fact_7897_fact__prod,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2))) ).

% fact_prod
tff(fact_7898_fact__prod,axiom,
    ! [N2: nat] : aa_nat_real(semiri2265585572941072030t_real,N2) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2))) ).

% fact_prod
tff(fact_7899_fact__prod,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2))) ).

% fact_prod
tff(fact_7900_fact__prod,axiom,
    ! [N2: nat] : aa_nat_complex(semiri5044797733671781792omplex,N2) = aa_nat_complex(semiri8010041392384452111omplex,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2))) ).

% fact_prod
tff(fact_7901_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,I2)),aa_complex_real(G,I2))) ) )
       => ( ( A2 != bot_bot_set_complex )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7902_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_Ext2541572767299617110t_real(F,I2)),aa_Ext2541572767299617110t_real(G,I2))) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups97031904164794029t_real(F,A2)),groups97031904164794029t_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7903_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( ! [I2: real] :
            ( member_real(I2,A2)
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,I2)),aa_real_real(G,I2))) ) )
       => ( ( A2 != bot_bot_set_real )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7904_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,I2)),aa_nat_real(G,I2))) ) )
       => ( ( A2 != bot_bot_set_nat )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A2)),aa_set_nat_real(groups129246275422532515t_real(G),A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7905_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),A2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,I2)),aa_int_real(G,I2))) ) )
       => ( ( A2 != bot_bot_set_int )
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7906_prod__mono__strict,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),A2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) ) )
       => ( ( A2 != bot_bot_set_complex )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7907_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),A2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_Extended_enat_rat(F,I2)),aa_Extended_enat_rat(G,I2))) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2245840878043517529at_rat(F,A2)),groups2245840878043517529at_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7908_prod__mono__strict,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( ! [I2: real] :
            ( member_real(I2,A2)
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) ) )
       => ( ( A2 != bot_bot_set_real )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7909_prod__mono__strict,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),A2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) ) )
       => ( ( A2 != bot_bot_set_nat )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups73079841787564623at_rat(F,A2)),groups73079841787564623at_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7910_prod__mono__strict,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),A2))
           => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)))
              & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) ) )
       => ( ( A2 != bot_bot_set_int )
         => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(G,A2))) ) ) ) ).

% prod_mono_strict
tff(fact_7911_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_7912_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_7913_binomial__maximum_H,axiom,
    ! [N2: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),N2))) ).

% binomial_maximum'
tff(fact_7914_binomial__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K6))) ) ) ).

% binomial_mono
tff(fact_7915_binomial__maximum,axiom,
    ! [N2: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% binomial_maximum
tff(fact_7916_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))) ) ) ) ).

% binomial_antimono
tff(fact_7917_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% binomial_le_pow2
tff(fact_7918_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_7919_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_7920_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_real,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7921_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_rat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7922_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_nat,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7923_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: fun_nat_int,P3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_7924_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa2: nat,Xb3: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa2),Xb3),Xc) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb3),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xb3),Xa2))
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa2),one_one_nat)),Xb3),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_7925_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [B: nat,A: nat,F: fun_nat_fun_nat_nat,Acc2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = Acc2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),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_7926_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_7927_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(gbinomial_real(A),K)) ).

% Suc_times_gbinomial
tff(fact_7928_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_7929_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_7930_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(gbinomial_real(A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_7931_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_7932_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_7933_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),M)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7934_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),M)),aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7935_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7936_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N2)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N2)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_7937_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_real] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B2,A2))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,B3))) )
         => ( ! [A3: real] :
                ( member_real(A3,A2)
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups1681761925125756287l_real(F,A2)),groups1681761925125756287l_real(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7938_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(F,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups2316167850115554303t_real(F,A2)),groups2316167850115554303t_real(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7939_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_complex_real(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups766887009212190081x_real(F,A2)),groups766887009212190081x_real(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7940_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B2))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2)))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,B3))) )
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,A3))) )
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups97031904164794029t_real(F,A2)),groups97031904164794029t_real(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7941_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_rat] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B2,A2))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B3))) )
         => ( ! [A3: real] :
                ( member_real(A3,A2)
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups4061424788464935467al_rat(F,A2)),groups4061424788464935467al_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7942_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: fun_int_rat] :
      ( pp(aa_set_int_bool(finite_finite_int,B2))
     => ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
       => ( ! [B3: int] :
              ( pp(aa_set_int_bool(member_int(B3),minus_minus_set_int(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B3))) )
         => ( ! [A3: int] :
                ( pp(aa_set_int_bool(member_int(A3),A2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A2)),groups1072433553688619179nt_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7943_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups225925009352817453ex_rat(F,A2)),groups225925009352817453ex_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7944_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,B2))
     => ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(A2),B2))
       => ( ! [B3: extended_enat] :
              ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(B3),minus_925952699566721837d_enat(B2,A2)))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_Extended_enat_rat(F,B3))) )
         => ( ! [A3: extended_enat] :
                ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(A3),A2))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,A3))) )
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2245840878043517529at_rat(F,A2)),groups2245840878043517529at_rat(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7945_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: fun_real_int] :
      ( finite_finite_real(B2)
     => ( ord_less_eq_set_real(A2,B2)
       => ( ! [B3: real] :
              ( member_real(B3,minus_minus_set_real(B2,A2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(F,B3))) )
         => ( ! [A3: real] :
                ( member_real(A3,A2)
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7946_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: fun_complex_int] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
     => ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
       => ( ! [B3: complex] :
              ( pp(aa_set_complex_bool(member_complex(B3),minus_811609699411566653omplex(B2,A2)))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_complex_int(F,B3))) )
         => ( ! [A3: complex] :
                ( pp(aa_set_complex_bool(member_complex(A3),A2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A3))) )
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(F,B2))) ) ) ) ) ).

% prod_mono2
tff(fact_7947_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_7948_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K6))) ) ) ).

% binomial_strict_mono
tff(fact_7949_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_7950_central__binomial__odd,axiom,
    ! [N2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_7951_gbinomial__code,axiom,
    ! [K: nat,A: rat] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = one_one_rat ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ok(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ) ) ).

% gbinomial_code
tff(fact_7952_gbinomial__code,axiom,
    ! [K: nat,A: real] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = one_one_real ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ol(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K)) ) ) ) ).

% gbinomial_code
tff(fact_7953_gbinomial__code,axiom,
    ! [K: nat,A: complex] :
      ( ( ( K = zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = one_one_complex ) )
      & ( ( K != zero_zero_nat )
       => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_om(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K)) ) ) ) ).

% gbinomial_code
tff(fact_7954_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_7955_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))) ) ) ).

% binomial_fact
tff(fact_7956_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))) ) ) ).

% binomial_fact
tff(fact_7957_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N2)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)))) ) ) ).

% binomial_fact
tff(fact_7958_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,N2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_binomial
tff(fact_7959_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,N2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_binomial
tff(fact_7960_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,N2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K))) ) ) ).

% fact_binomial
tff(fact_7961_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_7962_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(gbinomial_real(A),K)) ).

% gbinomial_factors
tff(fact_7963_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_7964_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_7965_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_7966_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_7967_gbinomial__negated__upper,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),A)),one_one_complex)),K)) ).

% gbinomial_negated_upper
tff(fact_7968_gbinomial__negated__upper,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_7969_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_7970_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N2))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N2)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),N2)) ).

% gbinomial_index_swap
tff(fact_7971_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N2))),one_one_real)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),N2)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),N2)) ).

% gbinomial_index_swap
tff(fact_7972_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),N2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),N2)) ).

% gbinomial_index_swap
tff(fact_7973_pochhammer__Suc__prod,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N2)) = groups6464643781859351333omplex(aTP_Lamm_on(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_7974_pochhammer__Suc__prod,axiom,
    ! [A: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_oo(A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_7975_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = groups73079841787564623at_rat(aTP_Lamm_op(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_7976_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_7977_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N2)) = groups705719431365010083at_int(aTP_Lamm_or(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod
tff(fact_7978_pochhammer__prod__rev,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),N2) = groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_os(A),N2),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_7979_pochhammer__prod__rev,axiom,
    ! [A: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),N2) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_ot(A),N2)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_7980_pochhammer__prod__rev,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),N2) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ou(A),N2),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_7981_pochhammer__prod__rev,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N2) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov(A),N2)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_7982_pochhammer__prod__rev,axiom,
    ! [A: int,N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),N2) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ow(A),N2),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2)) ).

% pochhammer_prod_rev
tff(fact_7983_binomial__code,axiom,
    ! [N2: nat,K: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),K))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),K)),one_one_nat)),N2),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_7984_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_7985_choose__two,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),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,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_7986_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_7987_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_7988_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_7989_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ox(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.in_pairs
tff(fact_7990_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,N2: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups73079841787564623at_rat(aTP_Lamm_oy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.in_pairs
tff(fact_7991_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_oz(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.in_pairs
tff(fact_7992_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = groups705719431365010083at_int(aTP_Lamm_pa(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ).

% prod.in_pairs
tff(fact_7993_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7994_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7995_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7996_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_pb(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7997_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_pc(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7998_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_pd(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7999_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_pe(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8000_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N2)) = groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_os(A),N2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8001_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(A),aa_nat_nat(suc,N2)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_ot(A),N2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8002_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N2)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ou(A),N2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8003_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov(A),N2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8004_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N2)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ow(A),N2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% pochhammer_Suc_prod_rev
tff(fact_8005_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_8006_gbinomial__pochhammer,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_8007_gbinomial__pochhammer,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer
tff(fact_8008_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_8009_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_8010_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_8011_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] : groups2073611262835488442omplex(aTP_Lamm_pf(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N2)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8012_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] : groups2906978787729119204at_rat(aTP_Lamm_pg(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8013_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ph,K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N2)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8014_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8015_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_real(gbinomial_real(A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8016_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8017_fact__code,axiom,
    ! [N2: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N2) = 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))),N2),one_one_nat)) ).

% fact_code
tff(fact_8018_fact__code,axiom,
    ! [N2: nat] : aa_nat_int(semiri1406184849735516958ct_int,N2) = 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))),N2),one_one_nat)) ).

% fact_code
tff(fact_8019_fact__code,axiom,
    ! [N2: nat] : aa_nat_real(semiri2265585572941072030t_real,N2) = 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))),N2),one_one_nat)) ).

% fact_code
tff(fact_8020_fact__code,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N2) = 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))),N2),one_one_nat)) ).

% fact_code
tff(fact_8021_fact__code,axiom,
    ! [N2: nat] : aa_nat_complex(semiri5044797733671781792omplex,N2) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2),one_one_nat)) ).

% fact_code
tff(fact_8022_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_md(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8023_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_me(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8024_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mf(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(gbinomial_real(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8025_choose__even__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_pi(N2),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_8026_choose__even__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_pj(N2),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_8027_choose__even__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( 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_pk,N2),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_8028_choose__even__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pl,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N2) ) ) ).

% choose_even_sum
tff(fact_8029_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aTP_Lamm_pm(N2),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_8030_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_pn(N2),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_8031_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( 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_po,N2),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_8032_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pp,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),N2) ) ) ).

% choose_odd_sum
tff(fact_8033_gbinomial__r__part__sum,axiom,
    ! [M: 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,M))),one_one_complex)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8034_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8035_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_8036_atMost__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_atMost_nat,X) = aa_nat_set_nat(set_ord_atMost_nat,Y) )
    <=> ( X = Y ) ) ).

% atMost_eq_iff
tff(fact_8037_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( member_real(I,set_ord_atMost_real(K))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),K)) ) ).

% atMost_iff
tff(fact_8038_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(I),K)) ) ).

% atMost_iff
tff(fact_8039_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( member_rat(I,set_ord_atMost_rat(K))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),K)) ) ).

% atMost_iff
tff(fact_8040_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_atMost_num(K))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),K)) ) ).

% atMost_iff
tff(fact_8041_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( pp(aa_set_int_bool(member_int(I),set_ord_atMost_int(K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K)) ) ).

% atMost_iff
tff(fact_8042_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).

% atMost_iff
tff(fact_8043_finite__atMost,axiom,
    ! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K))) ).

% finite_atMost
tff(fact_8044_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(X),set_or4236626031148496127et_nat(Y))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).

% atMost_subset_iff
tff(fact_8045_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(X),set_ord_atMost_rat(Y))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8046_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(X),set_ord_atMost_num(Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8047_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X)),set_ord_atMost_int(Y)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8048_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% atMost_subset_iff
tff(fact_8049_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( ( groups1707563613775114915nt_nat(F,A2) = one_one_nat )
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
           => ( aa_int_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8050_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( ( groups861055069439313189ex_nat(F,A2) = one_one_nat )
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
           => ( aa_complex_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8051_prod__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( ( groups4077766827762148844at_nat(F,A2) = one_one_nat )
      <=> ! [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
           => ( aa_Pro4243173770044397973at_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8052_prod__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( ( groups2880970938130013265at_nat(F,A2) = one_one_nat )
      <=> ! [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
           => ( aa_Extended_enat_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8053_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A2) = one_one_nat )
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
           => ( aa_nat_nat(F,X3) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8054_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H2: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H),set_or4236626031148496127et_nat(H2))
    <=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
        | pp(aa_set_nat_bool(ord_less_eq_set_nat(H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8055_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))
    <=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8056_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))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8057_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2)))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8058_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(H2)))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8059_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))
    <=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),H2)) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8060_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_8061_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_8062_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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,N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_8063_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) ).

% sum.atMost_Suc
tff(fact_8064_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_real(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_8065_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_rat(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_8066_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_nat(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_8067_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_nat_int(G,aa_nat_nat(suc,N2))) ).

% prod.atMost_Suc
tff(fact_8068_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( pp(aa_set_int_bool(finite_finite_int,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A2)))
      <=> ! [X3: int] :
            ( pp(aa_set_int_bool(member_int(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8069_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A2)))
      <=> ! [X3: complex] :
            ( pp(aa_set_complex_bool(member_complex(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8070_prod__pos__nat__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4077766827762148844at_nat(F,A2)))
      <=> ! [X3: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Pro4243173770044397973at_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8071_prod__pos__nat__iff,axiom,
    ! [A2: set_Extended_enat,F: fun_Ex7660298398204468809at_nat] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups2880970938130013265at_nat(F,A2)))
      <=> ! [X3: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_Extended_enat_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8072_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A2)))
      <=> ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),A2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8073_int__prod,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_ey(F),A2) ).

% int_prod
tff(fact_8074_int__prod,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A2)) = groups705719431365010083at_int(aTP_Lamm_fb(F),A2) ).

% int_prod
tff(fact_8075_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: extended_enat] : bot_bo7653980558646680370d_enat != set_or8332593352340944941d_enat(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8076_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: real] : bot_bot_set_real != set_ord_atMost_real(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8077_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] : bot_bot_set_int != set_ord_atMost_int(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8078_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8079_infinite__Iic,axiom,
    ! [A: int] : ~ pp(aa_set_int_bool(finite_finite_int,set_ord_atMost_int(A))) ).

% infinite_Iic
tff(fact_8080_not__Iic__eq__Icc,axiom,
    ! [H2: int,L: int,H: int] : set_ord_atMost_int(H2) != set_or1266510415728281911st_int(L,H) ).

% not_Iic_eq_Icc
tff(fact_8081_not__Iic__eq__Icc,axiom,
    ! [H2: real,L: real,H: real] : set_ord_atMost_real(H2) != set_or1222579329274155063t_real(L,H) ).

% not_Iic_eq_Icc
tff(fact_8082_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_pq,U)) ).

% atMost_def
tff(fact_8083_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_pr(U)) ).

% atMost_def
tff(fact_8084_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_ps,U)) ).

% atMost_def
tff(fact_8085_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_pt,U)) ).

% atMost_def
tff(fact_8086_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_pu,U)) ).

% atMost_def
tff(fact_8087_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_am,U)) ).

% atMost_def
tff(fact_8088_atMost__atLeast0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_ord_atMost_nat,N2) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2) ).

% atMost_atLeast0
tff(fact_8089_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_8090_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(H)),set_or1266510415728281911st_int(L2,H2))) ).

% not_Iic_le_Icc
tff(fact_8091_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ ord_less_eq_set_real(set_ord_atMost_real(H),set_or1222579329274155063t_real(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_8092_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(A),set_ord_lessThan_rat(B))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8093_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(A),set_ord_lessThan_num(B))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8094_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(A)),set_ord_lessThan_int(B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8095_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8096_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(set_ord_atMost_real(A),set_or5984915006950818249n_real(B))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8097_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = groups1705073143266064639nt_int(aTP_Lamm_av,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8098_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pv,M)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_8099_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_8100_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_8101_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = 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_hk(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_8102_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% sum.atMost_Suc_shift
tff(fact_8103_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_kk(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8104_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_kl(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_8105_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_km(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_8106_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,N2: nat,D: fun_nat_complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),X3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(D),X3),aa_nat_set_nat(set_ord_atMost_nat,N2))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N2))
         => ( aa_nat_complex(C,I3) = aa_nat_complex(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8107_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,N2: nat,D: fun_nat_real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(D),X3)),aa_nat_set_nat(set_ord_atMost_nat,N2))
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N2))
         => ( aa_nat_real(C,I3) = aa_nat_real(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8108_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B2: int] :
      ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N)))
     => ( ! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B2))
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_8109_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B2: nat] :
      ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N)))
     => ( ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B2))
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_8110_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B2: real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N)))
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B2))
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_8111_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8112_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8113_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8114_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% prod.atMost_Suc_shift
tff(fact_8115_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_av,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_8116_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_pw(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_py(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nested_swap'
tff(fact_8117_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pz(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qb(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% sum.nested_swap'
tff(fact_8118_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_qc(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qd(A),N2)),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nested_swap'
tff(fact_8119_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,N2: nat] : groups705719431365010083at_int(aTP_Lamm_qe(A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qg(A),N2),aa_nat_set_nat(set_ord_lessThan_nat,N2)) ).

% prod.nested_swap'
tff(fact_8120_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qh,R2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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),N2))),N2) ).

% sum_choose_lower
tff(fact_8121_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,N2)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8122_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,N2)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8123_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,N2: nat,K: nat] :
      ( ! [W: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),W),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_complex
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8124_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,N2: nat,K: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),W)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_real
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8125_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,N2: nat] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),X3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_complex
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N2))
         => ( aa_nat_complex(C,I3) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_8126_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,N2: nat] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_real
    <=> ! [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N2))
         => ( aa_nat_real(C,I3) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_8127_ln__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I2))) )
       => ( aa_real_real(ln_ln_real,groups1681761925125756287l_real(F,I5)) = groups8097168146408367636l_real(aTP_Lamm_qj(F),I5) ) ) ) ).

% ln_prod
tff(fact_8128_ln__prod,axiom,
    ! [I5: set_set_nat,F: fun_set_nat_real] :
      ( finite1152437895449049373et_nat(I5)
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I5)
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_set_nat_real(F,I2))) )
       => ( aa_real_real(ln_ln_real,groups3619160379726066777t_real(F,I5)) = groups5107569545109728110t_real(aTP_Lamm_qk(F),I5) ) ) ) ).

% ln_prod
tff(fact_8129_ln__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( pp(aa_set_int_bool(finite_finite_int,I5))
     => ( ! [I2: int] :
            ( pp(aa_set_int_bool(member_int(I2),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I2))) )
       => ( aa_real_real(ln_ln_real,groups2316167850115554303t_real(F,I5)) = groups8778361861064173332t_real(aTP_Lamm_ql(F),I5) ) ) ) ).

% ln_prod
tff(fact_8130_ln__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
     => ( ! [I2: complex] :
            ( pp(aa_set_complex_bool(member_complex(I2),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I2))) )
       => ( aa_real_real(ln_ln_real,groups766887009212190081x_real(F,I5)) = groups5808333547571424918x_real(aTP_Lamm_qm(F),I5) ) ) ) ).

% ln_prod
tff(fact_8131_ln__prod,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,I5))
     => ( ! [I2: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(I2),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Pro6168199777849092593t_real(F,I2))) )
       => ( aa_real_real(ln_ln_real,groups6036352826371341000t_real(F,I5)) = groups4567486121110086003t_real(aTP_Lamm_qn(F),I5) ) ) ) ).

% ln_prod
tff(fact_8132_ln__prod,axiom,
    ! [I5: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,I5))
     => ( ! [I2: extended_enat] :
            ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(I2),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I2))) )
       => ( aa_real_real(ln_ln_real,groups97031904164794029t_real(F,I5)) = groups4148127829035722712t_real(aTP_Lamm_qo(F),I5) ) ) ) ).

% ln_prod
tff(fact_8133_ln__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( pp(aa_set_nat_bool(finite_finite_nat,I5))
     => ( ! [I2: nat] :
            ( pp(aa_set_nat_bool(member_nat(I2),I5))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I2))) )
       => ( aa_real_real(ln_ln_real,aa_set_nat_real(groups129246275422532515t_real(F),I5)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qp(F)),I5) ) ) ) ).

% ln_prod
tff(fact_8134_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8135_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8136_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = 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_hk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8137_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% sum.atMost_shift
tff(fact_8138_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,M: nat,N2: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)))) ).

% sum_up_index_split
tff(fact_8139_sum__up__index__split,axiom,
    ! [F: fun_nat_int,M: nat,N2: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)))) ).

% sum_up_index_split
tff(fact_8140_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)))) ).

% sum_up_index_split
tff(fact_8141_sum__up__index__split,axiom,
    ! [F: fun_nat_real,M: nat,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)))) ).

% sum_up_index_split
tff(fact_8142_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cs(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8143_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,N2: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_ia(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8144_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_hk(G)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8145_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,N2: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_hn(G),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% prod.atMost_shift
tff(fact_8146_gbinomial__parallel__sum,axiom,
    ! [A: complex,N2: nat] : groups2073611262835488442omplex(aTP_Lamm_qq(A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N2))),one_one_complex)),N2) ).

% gbinomial_parallel_sum
tff(fact_8147_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] : groups2906978787729119204at_rat(aTP_Lamm_qr(A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,N2))),one_one_rat)),N2) ).

% gbinomial_parallel_sum
tff(fact_8148_gbinomial__parallel__sum,axiom,
    ! [A: real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qs(A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N2))),one_one_real)),N2) ).

% gbinomial_parallel_sum
tff(fact_8149_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qt(M),N2)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N2)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8150_vandermonde,axiom,
    ! [M: nat,N2: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qu(M),N2),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,M),N2)),R2) ).

% vandermonde
tff(fact_8151_sum__gp__basic,axiom,
    ! [X: complex,N2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_8152_sum__gp__basic,axiom,
    ! [X: rat,N2: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_8153_sum__gp__basic,axiom,
    ! [X: int,N2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_8154_sum__gp__basic,axiom,
    ! [X: real,N2: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N2))) ).

% sum_gp_basic
tff(fact_8155_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,N2: nat] :
      ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qv(C),N2))))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N2))
          & ( aa_nat_complex(C,I3) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_8156_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,N2: nat] :
      ( finite_finite_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_qw(C),N2)))
    <=> ? [I3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),N2))
          & ( aa_nat_real(C,I3) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_8157_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,N2: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_qv(C),N2)))) ) ) ).

% polyfun_roots_finite
tff(fact_8158_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,N2: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2))
       => finite_finite_real(collect_real(aa_nat_fun_real_bool(aTP_Lamm_qw(C),N2))) ) ) ).

% polyfun_roots_finite
tff(fact_8159_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,N2: nat] :
      ( ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ).

% polyfun_linear_factor_root
tff(fact_8160_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,N2: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(C),A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ).

% polyfun_linear_factor_root
tff(fact_8161_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,N2: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(C),A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ).

% polyfun_linear_factor_root
tff(fact_8162_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,N2: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ).

% polyfun_linear_factor_root
tff(fact_8163_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,N2: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N2)))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),A),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% polyfun_linear_factor
tff(fact_8164_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,N2: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N2)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(C),A),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% polyfun_linear_factor
tff(fact_8165_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,N2: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z3),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,N2)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(C),A),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% polyfun_linear_factor
tff(fact_8166_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,N2: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,N2)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),A)),aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% polyfun_linear_factor
tff(fact_8167_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_8168_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_8169_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_8170_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(X),M)),aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)))) ) ) ).

% sum_power_shift
tff(fact_8171_binomial,axiom,
    ! [A: nat,B: nat,N2: 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)),N2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qz(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial
tff(fact_8172_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N2: 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))),N2)))) = groups2906978787729119204at_rat(aTP_Lamm_iw(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_8173_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N2: 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))),N2)))) = groups3539618377306564664at_int(aTP_Lamm_ix(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_8174_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N2: 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))),N2)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_iy(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_8175_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_iz(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% sum.in_pairs_0
tff(fact_8176_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_complex,N2: nat,B: fun_nat_complex,X: complex] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_complex(A,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J2))
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(B),X),aa_nat_set_nat(set_ord_atMost_nat,N2))) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_rb(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ) ) ) ).

% polynomial_product
tff(fact_8177_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_rat,N2: nat,B: fun_nat_rat,X: rat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_rat(A,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J2))
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(B),X),aa_nat_set_nat(set_ord_atMost_nat,N2))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_rd(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ) ) ) ).

% polynomial_product
tff(fact_8178_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_int,N2: nat,B: fun_nat_int,X: int] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_int(A,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),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_qy(A),X),aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(B),X),aa_nat_set_nat(set_ord_atMost_nat,N2))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_rf(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ) ) ) ).

% polynomial_product
tff(fact_8179_polynomial__product,axiom,
    ! [M: nat,A: fun_nat_real,N2: nat,B: fun_nat_real,X: real] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_real(A,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J2))
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_rh(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ) ) ) ).

% polynomial_product
tff(fact_8180_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,N2: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ox(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8181_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,N2: 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))),N2)))) = groups73079841787564623at_rat(aTP_Lamm_oy(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8182_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_oz(G)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8183_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,N2: 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))),N2)))) = groups705719431365010083at_int(aTP_Lamm_pa(G),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% prod.in_pairs_0
tff(fact_8184_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,N2: nat,K: complex] :
      ( ! [X3: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),X3),aa_nat_set_nat(set_ord_atMost_nat,N2)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2)))
           => ( aa_nat_complex(C,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8185_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,N2: nat,K: real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2)))
           => ( aa_nat_real(C,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8186_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_ri(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8187_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_rj(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8188_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rk(A)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),M)),aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8189_binomial__ring,axiom,
    ! [A: complex,B: complex,N2: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),N2) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rl(A),B),N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_8190_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A),B)),N2) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rm(A),B),N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_8191_binomial__ring,axiom,
    ! [A: int,B: int,N2: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N2) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rn(A),B),N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_8192_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: 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)),N2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qz(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_8193_binomial__ring,axiom,
    ! [A: real,B: real,N2: nat] : aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),N2) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ro(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% binomial_ring
tff(fact_8194_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,N2: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),B)),N2) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rp(A),B),N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_8195_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),B)),N2) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rq(A),B),N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_8196_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),N2) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rr(A),B),N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_8197_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N2: nat] : aa_nat_real(comm_s7457072308508201937r_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),N2) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rs(A),B),N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) ).

% pochhammer_binomial_sum
tff(fact_8198_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,N2: nat,B: fun_nat_nat,X: nat] :
      ( ! [I2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),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_rt(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_rv(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2))) ) ) ) ).

% polynomial_product_nat
tff(fact_8199_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rw(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rx(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8200_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ry(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rz(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8201_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sa(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sb(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8202_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sc(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sd(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8203_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_se(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_sf(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8204_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sg(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sd(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8205_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sh(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rx(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8206_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_si(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sb(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8207_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),P3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),P3))
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sj(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rz(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8208_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_sk(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_sl(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8209_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_sm(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_sn(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8210_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_so(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_sp(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly
tff(fact_8211_root__polyfun,axiom,
    ! [N2: nat,Z2: int,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( ( aa_nat_int(power_power_int(Z2),N2) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_sq(N2),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_8212_root__polyfun,axiom,
    ! [N2: nat,Z2: complex,A: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( ( aa_nat_complex(power_power_complex(Z2),N2) = A )
      <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_sr(N2),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_8213_root__polyfun,axiom,
    ! [N2: nat,Z2: rat,A: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( ( aa_nat_rat(power_power_rat(Z2),N2) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ss(N2),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_8214_root__polyfun,axiom,
    ! [N2: nat,Z2: code_integer,A: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z2),N2) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_st(N2),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_8215_root__polyfun,axiom,
    ! [N2: nat,Z2: real,A: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( ( aa_nat_real(power_power_real(Z2),N2) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_su(N2),Z2),A)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_8216_sum__gp0,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)) ) )
      & ( ( X != one_one_complex )
       => ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N2)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).

% sum_gp0
tff(fact_8217_sum__gp0,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)) ) )
      & ( ( X != one_one_rat )
       => ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N2)))),aa_rat_rat(minus_minus_rat(one_one_rat),X)) ) ) ) ).

% sum_gp0
tff(fact_8218_sum__gp0,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)) ) )
      & ( ( X != one_one_real )
       => ( aa_set_nat_real(groups6591440286371151544t_real(power_power_real(X)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).

% sum_gp0
tff(fact_8219_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_sv(N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8220_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( groups2073611262835488442omplex(aTP_Lamm_sw(N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8221_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_sx(N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8222_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_sy,N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8223_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sz,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8224_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups2073611262835488442omplex(aTP_Lamm_ta(M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8225_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(aTP_Lamm_tb(M),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8226_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tc,M)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),M) ).

% gbinomial_sum_nat_pow2
tff(fact_8227_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] : groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_sk(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_td(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8228_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_sm(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_te(M),A),X),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8229_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_so(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_tf(M),A),X),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8230_polyfun__diff__alt,axiom,
    ! [N2: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(A),X),aa_nat_set_nat(set_ord_atMost_nat,N2))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_th(N2),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff_alt
tff(fact_8231_polyfun__diff__alt,axiom,
    ! [N2: nat,A: fun_nat_rat,X: rat,Y: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(A),X),aa_nat_set_nat(set_ord_atMost_nat,N2))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_tj(N2),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff_alt
tff(fact_8232_polyfun__diff__alt,axiom,
    ! [N2: nat,A: fun_nat_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(A),X),aa_nat_set_nat(set_ord_atMost_nat,N2))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_tl(N2),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff_alt
tff(fact_8233_polyfun__diff__alt,axiom,
    ! [N2: nat,A: fun_nat_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_tn(N2),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff_alt
tff(fact_8234_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_8235_choose__linear__sum,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_to,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_nat_nat(aa_nat_fun_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,N2),one_one_nat))) ).

% choose_linear_sum
tff(fact_8236_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( groups7501900531339628137nteger(aTP_Lamm_tp(N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8237_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( groups2073611262835488442omplex(aTP_Lamm_tq(N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_8238_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( groups2906978787729119204at_rat(aTP_Lamm_tr(N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_8239_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ts,N2),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8240_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tt,N2)),aa_nat_set_nat(set_ord_atMost_nat,N2)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8241_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: complex] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_complex_real(real_V1022390504157884413omplex,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(power_power_real(aa_complex_real(real_V1022390504157884413omplex,Z3)),aa_nat_nat(suc,N2))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8242_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
     => ? [M8: real] :
        ! [Z3: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,M8),aa_real_real(real_V7735802525324610683m_real,Z3)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(power_power_real(aa_real_real(real_V7735802525324610683m_real,Z3)),aa_nat_nat(suc,N2))))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8243_polyfun__diff,axiom,
    ! [N2: nat,A: fun_nat_complex,X: complex,Y: complex] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(A),X),aa_nat_set_nat(set_ord_atMost_nat,N2))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_tv(N2),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff
tff(fact_8244_polyfun__diff,axiom,
    ! [N2: nat,A: fun_nat_rat,X: rat,Y: rat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(A),X),aa_nat_set_nat(set_ord_atMost_nat,N2))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_tx(N2),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff
tff(fact_8245_polyfun__diff,axiom,
    ! [N2: nat,A: fun_nat_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(A),X),aa_nat_set_nat(set_ord_atMost_nat,N2))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_qy(A),Y),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_tz(N2),A),X),Y),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff
tff(fact_8246_polyfun__diff,axiom,
    ! [N2: nat,A: fun_nat_real,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(A),X)),aa_nat_set_nat(set_ord_atMost_nat,N2))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,N2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_ub(N2),A),X),Y)),aa_nat_set_nat(set_ord_lessThan_nat,N2))) ) ) ).

% polyfun_diff
tff(fact_8247_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ud(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y)))) ).

% sin_x_sin_y
tff(fact_8248_sin__x__sin__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_uf(X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))) ).

% sin_x_sin_y
tff(fact_8249_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_uh(X),Y)),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)))) ).

% sums_cos_x_plus_y
tff(fact_8250_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_uj(X),Y)),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))) ).

% sums_cos_x_plus_y
tff(fact_8251_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_8252_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z2) = one_one_real )
     => ~ ! [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))
             => ( Z2 != aa_real_complex(complex2(aa_real_real(cos_real,T2)),aa_real_real(sin_real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8253_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( aa_int_real(ring_1_of_int_real,W2) = aa_int_real(ring_1_of_int_real,Z2) )
    <=> ( W2 = Z2 ) ) ).

% of_int_eq_iff
tff(fact_8254_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,W2) = aa_int_rat(ring_1_of_int_rat,Z2) )
    <=> ( W2 = Z2 ) ) ).

% of_int_eq_iff
tff(fact_8255_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,W2) = aa_int_complex(ring_17405671764205052669omplex,Z2) )
    <=> ( W2 = Z2 ) ) ).

% of_int_eq_iff
tff(fact_8256_of__nat__id,axiom,
    ! [N2: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N2) = N2 ).

% of_nat_id
tff(fact_8257_scaleR__zero__right,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),zero_zero_real) = zero_zero_real ).

% scaleR_zero_right
tff(fact_8258_scaleR__cancel__right,axiom,
    ! [A: real,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X) )
    <=> ( ( A = B )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_cancel_right
tff(fact_8259_scaleR__eq__0__iff,axiom,
    ! [A: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
tff(fact_8260_scaleR__zero__left,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,zero_zero_real),X) = zero_zero_real ).

% scaleR_zero_left
tff(fact_8261_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_8262_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_8263_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_8264_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_8265_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z2) )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_8266_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z2) )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_8267_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z2) )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_8268_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_complex = aa_int_complex(ring_17405671764205052669omplex,Z2) )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_8269_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = zero_zero_int )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_8270_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = zero_zero_real )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_8271_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z2) = zero_zero_rat )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_8272_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z2) = zero_zero_complex )
    <=> ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_8273_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z2) = aa_num_rat(numeral_numeral_rat,N2) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,N2) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8274_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z2) = numera6690914467698888265omplex(N2) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,N2) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8275_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = aa_num_real(numeral_numeral_real,N2) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,N2) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8276_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = aa_num_int(numeral_numeral_int,N2) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,N2) ) ) ).

% of_int_eq_numeral_iff
tff(fact_8277_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_8278_of__int__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_8279_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_8280_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_8281_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ).

% of_int_le_iff
tff(fact_8282_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ).

% of_int_le_iff
tff(fact_8283_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ).

% of_int_le_iff
tff(fact_8284_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_8285_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% of_int_less_iff
tff(fact_8286_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% of_int_less_iff
tff(fact_8287_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ).

% of_int_less_iff
tff(fact_8288_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_8289_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_8290_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_8291_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_8292_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = one_one_int )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_8293_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = one_one_real )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_8294_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z2) = one_one_rat )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_8295_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z2) = one_one_complex )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_8296_of__int__mult,axiom,
    ! [W2: int,Z2: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)) = aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z2)) ).

% of_int_mult
tff(fact_8297_of__int__mult,axiom,
    ! [W2: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_mult
tff(fact_8298_of__int__mult,axiom,
    ! [W2: int,Z2: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% of_int_mult
tff(fact_8299_of__int__mult,axiom,
    ! [W2: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_mult
tff(fact_8300_of__int__add,axiom,
    ! [W2: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_add
tff(fact_8301_of__int__add,axiom,
    ! [W2: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_add
tff(fact_8302_of__int__add,axiom,
    ! [W2: int,Z2: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% of_int_add
tff(fact_8303_of__int__add,axiom,
    ! [W2: int,Z2: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),Z2)) = aa_complex_complex(plus_plus_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z2)) ).

% of_int_add
tff(fact_8304_of__int__minus,axiom,
    ! [Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,Z2)) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_minus
tff(fact_8305_of__int__minus,axiom,
    ! [Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,Z2)) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_minus
tff(fact_8306_of__int__minus,axiom,
    ! [Z2: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,Z2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,Z2)) ).

% of_int_minus
tff(fact_8307_of__int__minus,axiom,
    ! [Z2: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z2)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z2)) ).

% of_int_minus
tff(fact_8308_of__int__minus,axiom,
    ! [Z2: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,Z2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z2)) ).

% of_int_minus
tff(fact_8309_of__int__diff,axiom,
    ! [W2: int,Z2: int] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z2)) = aa_complex_complex(minus_minus_complex(aa_int_complex(ring_17405671764205052669omplex,W2)),aa_int_complex(ring_17405671764205052669omplex,Z2)) ).

% of_int_diff
tff(fact_8310_of__int__diff,axiom,
    ! [W2: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_diff
tff(fact_8311_of__int__diff,axiom,
    ! [W2: int,Z2: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z2)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% of_int_diff
tff(fact_8312_of__int__diff,axiom,
    ! [W2: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W2),Z2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_diff
tff(fact_8313_of__int__of__nat__eq,axiom,
    ! [N2: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_complex(semiri8010041392384452111omplex,N2) ).

% of_int_of_nat_eq
tff(fact_8314_of__int__of__nat__eq,axiom,
    ! [N2: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_real(semiri5074537144036343181t_real,N2) ).

% of_int_of_nat_eq
tff(fact_8315_of__int__of__nat__eq,axiom,
    ! [N2: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_rat(semiri681578069525770553at_rat,N2) ).

% of_int_of_nat_eq
tff(fact_8316_of__int__of__nat__eq,axiom,
    ! [N2: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_int(semiri1314217659103216013at_int,N2) ).

% of_int_of_nat_eq
tff(fact_8317_of__int__abs,axiom,
    ! [X: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,X)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,X)) ).

% of_int_abs
tff(fact_8318_of__int__abs,axiom,
    ! [X: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,X)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,X)) ).

% of_int_abs
tff(fact_8319_of__int__abs,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,X)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,X)) ).

% of_int_abs
tff(fact_8320_of__int__abs,axiom,
    ! [X: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,X)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,X)) ).

% of_int_abs
tff(fact_8321_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8322_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8323_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8324_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) )
    <=> ( X = aa_nat_int(power_power_int(B),W2) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_8325_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2) = aa_int_rat(ring_1_of_int_rat,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8326_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8327_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8328_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W2) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(power_power_int(B),W2) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_8329_of__int__power,axiom,
    ! [Z2: int,N2: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z2),N2)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z2)),N2) ).

% of_int_power
tff(fact_8330_of__int__power,axiom,
    ! [Z2: int,N2: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z2),N2)) = aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,Z2)),N2) ).

% of_int_power
tff(fact_8331_of__int__power,axiom,
    ! [Z2: int,N2: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z2),N2)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z2)),N2) ).

% of_int_power
tff(fact_8332_of__int__power,axiom,
    ! [Z2: int,N2: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z2),N2)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z2)),N2) ).

% of_int_power
tff(fact_8333_frac__of__int,axiom,
    ! [Z2: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z2)) = zero_zero_real ).

% frac_of_int
tff(fact_8334_frac__of__int,axiom,
    ! [Z2: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z2)) = zero_zero_rat ).

% frac_of_int
tff(fact_8335_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_8336_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_8337_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_8338_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A2)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_uk(F)),A2) ).

% of_int_prod
tff(fact_8339_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A2)) = groups73079841787564623at_rat(aTP_Lamm_ul(F),A2) ).

% of_int_prod
tff(fact_8340_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_complex(ring_17405671764205052669omplex,groups705719431365010083at_int(F,A2)) = groups6464643781859351333omplex(aTP_Lamm_um(F),A2) ).

% of_int_prod
tff(fact_8341_of__int__prod,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A2)) = groups705719431365010083at_int(aTP_Lamm_un(F),A2) ).

% of_int_prod
tff(fact_8342_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A2)) = groups2316167850115554303t_real(aTP_Lamm_uo(F),A2) ).

% of_int_prod
tff(fact_8343_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A2)) = groups1072433553688619179nt_rat(aTP_Lamm_up(F),A2) ).

% of_int_prod
tff(fact_8344_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups1705073143266064639nt_int(F,A2)) = groups7440179247065528705omplex(aTP_Lamm_uq(F),A2) ).

% of_int_prod
tff(fact_8345_of__int__prod,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A2)) = groups1705073143266064639nt_int(aTP_Lamm_ur(F),A2) ).

% of_int_prod
tff(fact_8346_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_8347_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uk(F)),A2) ).

% of_int_sum
tff(fact_8348_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_us(F),A2) ).

% of_int_sum
tff(fact_8349_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_uo(F),A2) ).

% of_int_sum
tff(fact_8350_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_up(F),A2) ).

% of_int_sum
tff(fact_8351_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_complex(ring_17405671764205052669omplex,groups4538972089207619220nt_int(F,A2)) = groups3049146728041665814omplex(aTP_Lamm_uq(F),A2) ).

% of_int_sum
tff(fact_8352_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_ur(F),A2) ).

% of_int_sum
tff(fact_8353_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) ) ).

% of_int_0_le_iff
tff(fact_8354_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) ) ).

% of_int_0_le_iff
tff(fact_8355_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) ) ).

% of_int_0_le_iff
tff(fact_8356_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_8357_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_8358_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int)) ) ).

% of_int_le_0_iff
tff(fact_8359_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% of_int_0_less_iff
tff(fact_8360_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% of_int_0_less_iff
tff(fact_8361_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% of_int_0_less_iff
tff(fact_8362_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),zero_zero_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_8363_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),zero_zero_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_8364_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),zero_zero_int)) ) ).

% of_int_less_0_iff
tff(fact_8365_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,N2)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z2)) ) ).

% of_int_numeral_le_iff
tff(fact_8366_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z2)) ) ).

% of_int_numeral_le_iff
tff(fact_8367_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N2)),Z2)) ) ).

% of_int_numeral_le_iff
tff(fact_8368_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),aa_num_real(numeral_numeral_real,N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),aa_num_int(numeral_numeral_int,N2))) ) ).

% of_int_le_numeral_iff
tff(fact_8369_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),aa_num_rat(numeral_numeral_rat,N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),aa_num_int(numeral_numeral_int,N2))) ) ).

% of_int_le_numeral_iff
tff(fact_8370_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),aa_num_int(numeral_numeral_int,N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),aa_num_int(numeral_numeral_int,N2))) ) ).

% of_int_le_numeral_iff
tff(fact_8371_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),aa_num_rat(numeral_numeral_rat,N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_num_int(numeral_numeral_int,N2))) ) ).

% of_int_less_numeral_iff
tff(fact_8372_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),aa_num_real(numeral_numeral_real,N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_num_int(numeral_numeral_int,N2))) ) ).

% of_int_less_numeral_iff
tff(fact_8373_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),aa_num_int(numeral_numeral_int,N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),aa_num_int(numeral_numeral_int,N2))) ) ).

% of_int_less_numeral_iff
tff(fact_8374_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_num_rat(numeral_numeral_rat,N2)),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z2)) ) ).

% of_int_numeral_less_iff
tff(fact_8375_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_num_real(numeral_numeral_real,N2)),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z2)) ) ).

% of_int_numeral_less_iff
tff(fact_8376_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N2)),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N2)),Z2)) ) ).

% of_int_numeral_less_iff
tff(fact_8377_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_8378_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_8379_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),one_one_int)) ) ).

% of_int_le_1_iff
tff(fact_8380_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z2)) ) ).

% of_int_1_le_iff
tff(fact_8381_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z2)) ) ).

% of_int_1_le_iff
tff(fact_8382_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z2)) ) ).

% of_int_1_le_iff
tff(fact_8383_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),one_one_real))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_8384_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_8385_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),one_one_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),one_one_int)) ) ).

% of_int_less_1_iff
tff(fact_8386_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z2)) ) ).

% of_int_1_less_iff
tff(fact_8387_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z2)) ) ).

% of_int_1_less_iff
tff(fact_8388_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z2)) ) ).

% of_int_1_less_iff
tff(fact_8389_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8390_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N2) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8391_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N2) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8392_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_8393_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8394_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8395_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8396_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_8397_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_8398_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_8399_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_8400_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_8401_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_8402_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_8403_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_8404_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_8405_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W2)),X)) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_8406_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(power_power_real(aa_int_real(ring_1_of_int_real,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_8407_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_8408_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W2))) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_8409_sin__npi__int,axiom,
    ! [N2: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N2))) = zero_zero_real ).

% sin_npi_int
tff(fact_8410_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),aa_num_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_8411_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_8412_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_8413_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_8414_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N2)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_8415_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_8416_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_8417_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_8418_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N2)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_8419_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_8420_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_8421_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_8422_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_8423_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8424_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8425_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8426_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8427_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_8428_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8429_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N2) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8430_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N2) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8431_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8432_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N2) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_8433_sin__int__2pin,axiom,
    ! [N2: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N2))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8434_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8435_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8436_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8437_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_8438_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N2)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8439_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N2)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8440_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8441_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_8442_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),aa_int_int(ring_1_of_int_int,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8443_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N2)),aa_int_real(ring_1_of_int_real,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8444_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2)),aa_int_rat(ring_1_of_int_rat,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8445_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N2)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)),A)) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_8446_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8447_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8448_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8449_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N2))) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_8450_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% ex_le_of_int
tff(fact_8451_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ).

% ex_le_of_int
tff(fact_8452_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ).

% ex_of_int_less
tff(fact_8453_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ).

% ex_of_int_less
tff(fact_8454_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z))) ).

% ex_less_of_int
tff(fact_8455_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ).

% ex_less_of_int
tff(fact_8456_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] : aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_int_complex(ring_17405671764205052669omplex,X)) ).

% mult_of_int_commute
tff(fact_8457_mult__of__int__commute,axiom,
    ! [X: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_8458_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X)) ).

% mult_of_int_commute
tff(fact_8459_mult__of__int__commute,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_8460_scaleR__right__imp__eq,axiom,
    ! [X: real,A: real,B: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
tff(fact_8461_scaleR__right__distrib,axiom,
    ! [A: real,X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) ).

% scaleR_right_distrib
tff(fact_8462_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,Y)) ).

% of_int_max
tff(fact_8463_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_rat_rat(ord_max_rat(aa_int_rat(ring_1_of_int_rat,X)),aa_int_rat(ring_1_of_int_rat,Y)) ).

% of_int_max
tff(fact_8464_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,X)),aa_int_int(ring_1_of_int_int,Y)) ).

% of_int_max
tff(fact_8465_of__int__max,axiom,
    ! [X: int,Y: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)) = ord_max_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,X),aa_int_Code_integer(ring_18347121197199848620nteger,Y)) ).

% of_int_max
tff(fact_8466_Ints__of__int,axiom,
    ! [Z2: int] : pp(aa_set_int_bool(member_int(aa_int_int(ring_1_of_int_int,Z2)),ring_1_Ints_int)) ).

% Ints_of_int
tff(fact_8467_Ints__of__int,axiom,
    ! [Z2: int] : member_real(aa_int_real(ring_1_of_int_real,Z2),ring_1_Ints_real) ).

% Ints_of_int
tff(fact_8468_Ints__of__int,axiom,
    ! [Z2: int] : member_rat(aa_int_rat(ring_1_of_int_rat,Z2),ring_1_Ints_rat) ).

% Ints_of_int
tff(fact_8469_Ints__of__int,axiom,
    ! [Z2: int] : pp(aa_set_complex_bool(member_complex(aa_int_complex(ring_17405671764205052669omplex,Z2)),ring_1_Ints_complex)) ).

% Ints_of_int
tff(fact_8470_Ints__induct,axiom,
    ! [Q4: int,P: fun_int_bool] :
      ( pp(aa_set_int_bool(member_int(Q4),ring_1_Ints_int))
     => ( ! [Z: int] : pp(aa_int_bool(P,aa_int_int(ring_1_of_int_int,Z)))
       => pp(aa_int_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_8471_Ints__induct,axiom,
    ! [Q4: real,P: fun_real_bool] :
      ( member_real(Q4,ring_1_Ints_real)
     => ( ! [Z: int] : pp(aa_real_bool(P,aa_int_real(ring_1_of_int_real,Z)))
       => pp(aa_real_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_8472_Ints__induct,axiom,
    ! [Q4: rat,P: fun_rat_bool] :
      ( member_rat(Q4,ring_1_Ints_rat)
     => ( ! [Z: int] : pp(aa_rat_bool(P,aa_int_rat(ring_1_of_int_rat,Z)))
       => pp(aa_rat_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_8473_Ints__induct,axiom,
    ! [Q4: complex,P: fun_complex_bool] :
      ( pp(aa_set_complex_bool(member_complex(Q4),ring_1_Ints_complex))
     => ( ! [Z: int] : pp(aa_complex_bool(P,aa_int_complex(ring_17405671764205052669omplex,Z)))
       => pp(aa_complex_bool(P,Q4)) ) ) ).

% Ints_induct
tff(fact_8474_Ints__cases,axiom,
    ! [Q4: int] :
      ( pp(aa_set_int_bool(member_int(Q4),ring_1_Ints_int))
     => ~ ! [Z: int] : Q4 != aa_int_int(ring_1_of_int_int,Z) ) ).

% Ints_cases
tff(fact_8475_Ints__cases,axiom,
    ! [Q4: real] :
      ( member_real(Q4,ring_1_Ints_real)
     => ~ ! [Z: int] : Q4 != aa_int_real(ring_1_of_int_real,Z) ) ).

% Ints_cases
tff(fact_8476_Ints__cases,axiom,
    ! [Q4: rat] :
      ( member_rat(Q4,ring_1_Ints_rat)
     => ~ ! [Z: int] : Q4 != aa_int_rat(ring_1_of_int_rat,Z) ) ).

% Ints_cases
tff(fact_8477_Ints__cases,axiom,
    ! [Q4: complex] :
      ( pp(aa_set_complex_bool(member_complex(Q4),ring_1_Ints_complex))
     => ~ ! [Z: int] : Q4 != aa_int_complex(ring_17405671764205052669omplex,Z) ) ).

% Ints_cases
tff(fact_8478_scale__right__distrib__NO__MATCH,axiom,
    ! [X: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(nO_MATCH_real_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_8479_scale__right__distrib__NO__MATCH,axiom,
    ! [X: complex,Y: complex,A: real] :
      ( pp(aa_real_bool(nO_MAT9165723751696580935x_real(aa_complex_complex(divide1717551699836669952omplex(X),Y)),A))
     => ( aa_complex_complex(real_V2046097035970521341omplex(A),aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(real_V2046097035970521341omplex(A),X)),aa_complex_complex(real_V2046097035970521341omplex(A),Y)) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_8480_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X)) ).

% scaleR_left_distrib
tff(fact_8481_scaleR__left_Oadd,axiom,
    ! [X: real,Y: real,Xa2: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),Xa2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,X),Xa2)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,Y),Xa2)) ).

% scaleR_left.add
tff(fact_8482_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W2) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W2) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_8483_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
tff(fact_8484_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_8485_scaleR__right__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),X))) ) ) ).

% scaleR_right_mono
tff(fact_8486_scaleR__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),C))) ) ) ).

% scaleR_right_mono_neg
tff(fact_8487_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) = X )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,V),X) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_8488_eq__vector__fraction__iff,axiom,
    ! [X: real,U: real,V: real,A: real] :
      ( ( X = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),V)),A) )
    <=> ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,V),X) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,U),A) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_8489_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E2)),C)),D)) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_8490_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),E2)),D)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E2)),D))) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_8491_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_8492_real__of__int__div4,axiom,
    ! [N2: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N2),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N2)),aa_int_real(ring_1_of_int_real,X)))) ).

% real_of_int_div4
tff(fact_8493_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => member_rat(aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A)),aa_int_rat(ring_1_of_int_rat,B)),ring_1_Ints_rat) ) ).

% of_int_divide_in_Ints
tff(fact_8494_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => pp(aa_set_int_bool(member_int(aa_int_int(divide_divide_int(aa_int_int(ring_1_of_int_int,A)),aa_int_int(ring_1_of_int_int,B))),ring_1_Ints_int)) ) ).

% of_int_divide_in_Ints
tff(fact_8495_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => member_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B)),ring_1_Ints_real) ) ).

% of_int_divide_in_Ints
tff(fact_8496_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
     => pp(aa_set_complex_bool(member_complex(aa_complex_complex(divide1717551699836669952omplex(aa_int_complex(ring_17405671764205052669omplex,A)),aa_int_complex(ring_17405671764205052669omplex,B))),ring_1_Ints_complex)) ) ).

% of_int_divide_in_Ints
tff(fact_8497_real__of__int__div,axiom,
    ! [D: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),N2))
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N2),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N2)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_8498_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_8499_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_8500_eval__nat__numeral_I2_J,axiom,
    ! [N2: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N2)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N2))) ).

% eval_nat_numeral(2)
tff(fact_8501_one__plus__BitM,axiom,
    ! [N2: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,N2)) = aa_num_num(bit0,N2) ).

% one_plus_BitM
tff(fact_8502_BitM__plus__one,axiom,
    ! [N2: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,N2)),one) = aa_num_num(bit0,N2) ).

% BitM_plus_one
tff(fact_8503_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2))) ) ).

% of_int_nonneg
tff(fact_8504_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2))) ) ).

% of_int_nonneg
tff(fact_8505_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2))) ) ).

% of_int_nonneg
tff(fact_8506_of__int__leD,axiom,
    ! [N2: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),X)) ) ) ).

% of_int_leD
tff(fact_8507_of__int__leD,axiom,
    ! [N2: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).

% of_int_leD
tff(fact_8508_of__int__leD,axiom,
    ! [N2: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).

% of_int_leD
tff(fact_8509_of__int__leD,axiom,
    ! [N2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).

% of_int_leD
tff(fact_8510_of__int__pos,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2))) ) ).

% of_int_pos
tff(fact_8511_of__int__pos,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2))) ) ).

% of_int_pos
tff(fact_8512_of__int__pos,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2))) ) ).

% of_int_pos
tff(fact_8513_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)),zero_zero_real))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
tff(fact_8514_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B)))
    <=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
tff(fact_8515_of__int__lessD,axiom,
    ! [N2: int,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X)) ) ) ).

% of_int_lessD
tff(fact_8516_of__int__lessD,axiom,
    ! [N2: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).

% of_int_lessD
tff(fact_8517_of__int__lessD,axiom,
    ! [N2: int,X: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).

% of_int_lessD
tff(fact_8518_of__int__lessD,axiom,
    ! [N2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N2))),X))
     => ( ( N2 = zero_zero_int )
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).

% of_int_lessD
tff(fact_8519_scaleR__mono,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Y))) ) ) ) ) ).

% scaleR_mono
tff(fact_8520_scaleR__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),D))) ) ) ) ) ).

% scaleR_mono'
tff(fact_8521_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real)) ) ).

% split_scaleR_neg_le
tff(fact_8522_split__scaleR__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
        | ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B))) ) ).

% split_scaleR_pos_le
tff(fact_8523_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X))) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_8524_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real)) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_8525_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),zero_zero_real)) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_8526_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),B))) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_8527_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)),X)) ) ) ).

% scaleR_left_le_one_le
tff(fact_8528_floor__exists,axiom,
    ! [X: real] :
    ? [Z: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))) ) ).

% floor_exists
tff(fact_8529_floor__exists,axiom,
    ! [X: rat] :
    ? [Z: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))) ) ).

% floor_exists
tff(fact_8530_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_8531_floor__exists1,axiom,
    ! [X: rat] :
    ? [X4: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X4)),X))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int))))
      & ! [Y4: int] :
          ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),X))
            & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int)))) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
tff(fact_8532_scaleR__2,axiom,
    ! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X) ).

% scaleR_2
tff(fact_8533_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_8534_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_8535_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_8536_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_8537_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_8538_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_int_real(ring_1_of_int_real,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_8539_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),aa_int_rat(ring_1_of_int_rat,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_8540_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_int_int(ring_1_of_int_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N2)),X)) ) ).

% of_nat_less_of_int_iff
tff(fact_8541_Complex__sum_H,axiom,
    ! [F: fun_nat_real,S: set_nat] : groups2073611262835488442omplex(aTP_Lamm_ut(F),S) = aa_real_complex(complex2(aa_set_nat_real(groups6591440286371151544t_real(F),S)),zero_zero_real) ).

% Complex_sum'
tff(fact_8542_Complex__sum_H,axiom,
    ! [F: fun_complex_real,S: set_complex] : groups7754918857620584856omplex(aTP_Lamm_uu(F),S) = aa_real_complex(complex2(groups5808333547571424918x_real(F,S)),zero_zero_real) ).

% Complex_sum'
tff(fact_8543_int__le__real__less,axiom,
    ! [N2: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N2),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real))) ) ).

% int_le_real_less
tff(fact_8544_int__less__real__le,axiom,
    ! [N2: int,M: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N2),M))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N2)),one_one_real)),aa_int_real(ring_1_of_int_real,M))) ) ).

% int_less_real_le
tff(fact_8545_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I3: int] : X = 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_8546_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(X),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_8547_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_8548_numeral__BitM,axiom,
    ! [N2: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,N2)) = aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,N2))),one_one_rat) ).

% numeral_BitM
tff(fact_8549_numeral__BitM,axiom,
    ! [N2: num] : numera6690914467698888265omplex(aa_num_num(bitM,N2)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,N2))),one_one_complex) ).

% numeral_BitM
tff(fact_8550_numeral__BitM,axiom,
    ! [N2: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,N2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,N2))),one_one_real) ).

% numeral_BitM
tff(fact_8551_numeral__BitM,axiom,
    ! [N2: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,N2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2))),one_one_int) ).

% numeral_BitM
tff(fact_8552_real__of__int__div2,axiom,
    ! [N2: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N2)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N2),X))))) ).

% real_of_int_div2
tff(fact_8553_real__of__int__div3,axiom,
    ! [N2: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N2)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(N2),X)))),one_one_real)) ).

% real_of_int_div3
tff(fact_8554_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I3: int] :
          ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_8555_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I3))
          & ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I3)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8556_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_uw(X),Y)),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y)))) ).

% cos_x_cos_y
tff(fact_8557_cos__x__cos__y,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_uy(X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))) ).

% cos_x_cos_y
tff(fact_8558_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8559_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8560_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8561_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_rat(ring_1_of_int_rat,K) = zero_zero_rat ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_rat(ring_1_of_int_rat,K) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_rat)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8562_of__int__code__if,axiom,
    ! [K: int] :
      ( ( ( K = zero_zero_int )
       => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))) ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
           => ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% of_int_code_if
tff(fact_8563_round__unique,axiom,
    ! [X: real,Y: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).

% round_unique
tff(fact_8564_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y ) ) ) ).

% round_unique
tff(fact_8565_round__unique_H,axiom,
    ! [X: rat,N2: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,N2)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = N2 ) ) ).

% round_unique'
tff(fact_8566_round__unique_H,axiom,
    ! [X: real,N2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,N2)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = N2 ) ) ).

% round_unique'
tff(fact_8567_of__int__round__abs__le,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_8568_of__int__round__abs__le,axiom,
    ! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_abs_le
tff(fact_8569_exp__first__two__terms,axiom,
    ! [X: real] : aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),suminf_real(aTP_Lamm_uz(X))) ).

% exp_first_two_terms
tff(fact_8570_exp__first__two__terms,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),X)),suminf_complex(aTP_Lamm_va(X))) ).

% exp_first_two_terms
tff(fact_8571_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_8572_inverse__eq__iff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_8573_inverse__inverse__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ).

% inverse_inverse_eq
tff(fact_8574_inverse__inverse__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ).

% inverse_inverse_eq
tff(fact_8575_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8576_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8577_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_8578_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_8579_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_8580_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).

% inverse_mult_distrib
tff(fact_8581_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_8582_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_8583_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_8584_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,X) = one_one_complex )
    <=> ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_8585_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(inverse_inverse_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_8586_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,X) = one_one_rat )
    <=> ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_8587_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_8588_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_8589_inverse__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(B),A) ).

% inverse_divide
tff(fact_8590_inverse__minus__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ).

% inverse_minus_eq
tff(fact_8591_inverse__minus__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ).

% inverse_minus_eq
tff(fact_8592_inverse__minus__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ).

% inverse_minus_eq
tff(fact_8593_abs__inverse,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ).

% abs_inverse
tff(fact_8594_abs__inverse,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ).

% abs_inverse
tff(fact_8595_round__of__int,axiom,
    ! [N2: int] : aa_real_int(archim8280529875227126926d_real,aa_int_real(ring_1_of_int_real,N2)) = N2 ).

% round_of_int
tff(fact_8596_round__of__int,axiom,
    ! [N2: int] : aa_rat_int(archim7778729529865785530nd_rat,aa_int_rat(ring_1_of_int_rat,N2)) = N2 ).

% round_of_int
tff(fact_8597_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8598_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8599_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8600_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8601_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).

% inverse_positive_iff_positive
tff(fact_8602_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).

% inverse_positive_iff_positive
tff(fact_8603_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).

% inverse_negative_iff_negative
tff(fact_8604_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).

% inverse_negative_iff_negative
tff(fact_8605_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8606_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8607_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).

% inverse_less_iff_less
tff(fact_8608_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).

% inverse_less_iff_less
tff(fact_8609_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_8610_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_8611_round__numeral,axiom,
    ! [N2: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,N2)) = aa_num_int(numeral_numeral_int,N2) ).

% 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,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_8614_round__of__nat,axiom,
    ! [N2: nat] : aa_real_int(archim8280529875227126926d_real,aa_nat_real(semiri5074537144036343181t_real,N2)) = aa_nat_int(semiri1314217659103216013at_int,N2) ).

% round_of_nat
tff(fact_8615_round__of__nat,axiom,
    ! [N2: nat] : aa_rat_int(archim7778729529865785530nd_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)) = aa_nat_int(semiri1314217659103216013at_int,N2) ).

% round_of_nat
tff(fact_8616_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8617_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8618_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).

% inverse_le_iff_le
tff(fact_8619_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
        <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).

% inverse_le_iff_le
tff(fact_8620_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_8621_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_8622_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_8623_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_8624_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_8625_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_8626_inverse__eq__divide__numeral,axiom,
    ! [W2: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W2)) ).

% inverse_eq_divide_numeral
tff(fact_8627_inverse__eq__divide__numeral,axiom,
    ! [W2: num] : aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,W2)) ).

% inverse_eq_divide_numeral
tff(fact_8628_inverse__eq__divide__numeral,axiom,
    ! [W2: num] : aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,W2)) ).

% inverse_eq_divide_numeral
tff(fact_8629_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W2))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8630_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8631_inverse__eq__divide__neg__numeral,axiom,
    ! [W2: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8632_round__neg__numeral,axiom,
    ! [N2: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) ).

% round_neg_numeral
tff(fact_8633_round__neg__numeral,axiom,
    ! [N2: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,N2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2)) ).

% round_neg_numeral
tff(fact_8634_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(real_V7735802525324610683m_real,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_8635_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_real_real(inverse_inverse_real,aa_complex_real(real_V1022390504157884413omplex,A)) ) ) ).

% nonzero_norm_inverse
tff(fact_8636_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_8637_inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_8638_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Y)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8639_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,X: rat] :
      ( ( aa_rat_rat(times_times_rat(Y),X) = aa_rat_rat(times_times_rat(X),Y) )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Y)),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8640_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_8641_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_8642_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_8643_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_8644_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8645_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8646_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8647_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8648_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8649_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8650_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,X)) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_8651_norm__inverse__le__norm,axiom,
    ! [R2: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(real_V7735802525324610683m_real,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,X))),aa_real_real(inverse_inverse_real,R2))) ) ) ).

% norm_inverse_le_norm
tff(fact_8652_norm__inverse__le__norm,axiom,
    ! [R2: real,X: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_complex_real(real_V1022390504157884413omplex,X)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,X))),aa_real_real(inverse_inverse_real,R2))) ) ) ).

% norm_inverse_le_norm
tff(fact_8653_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))) ) ).

% positive_imp_inverse_positive
tff(fact_8654_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))) ) ).

% positive_imp_inverse_positive
tff(fact_8655_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)) ) ).

% negative_imp_inverse_negative
tff(fact_8656_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
     => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)) ) ).

% negative_imp_inverse_negative
tff(fact_8657_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)))
     => ( ( A != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ).

% inverse_positive_imp_positive
tff(fact_8658_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)))
     => ( ( A != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ).

% inverse_positive_imp_positive
tff(fact_8659_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real))
     => ( ( A != zero_zero_real )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).

% inverse_negative_imp_negative
tff(fact_8660_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat))
     => ( ( A != zero_zero_rat )
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).

% inverse_negative_imp_negative
tff(fact_8661_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8662_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8663_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8664_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8665_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A))) ) ) ).

% less_imp_inverse_less
tff(fact_8666_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A))) ) ) ).

% less_imp_inverse_less
tff(fact_8667_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).

% inverse_less_imp_less
tff(fact_8668_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
       => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).

% inverse_less_imp_less
tff(fact_8669_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_8670_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8671_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_8672_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,one)) = aa_num_real(numeral_numeral_real,one) ).

% inverse_numeral_1
tff(fact_8673_inverse__numeral__1,axiom,
    aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_num_rat(numeral_numeral_rat,one) ).

% inverse_numeral_1
tff(fact_8674_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8675_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8676_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8677_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_8678_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_8679_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = one_one_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).

% inverse_unique
tff(fact_8680_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_8681_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_8682_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% field_class.field_divide_inverse
tff(fact_8683_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_8684_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_8685_divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% divide_inverse
tff(fact_8686_divide__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_8687_forall__pos__mono__1,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D5: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D5),E))
         => ( pp(aa_real_bool(P,D5))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_8688_forall__pos__mono,axiom,
    ! [P: fun_real_bool,E2: real] :
      ( ! [D5: real,E: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D5),E))
         => ( pp(aa_real_bool(P,D5))
           => pp(aa_real_bool(P,E)) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N)))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
         => pp(aa_real_bool(P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_8689_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E2)) ) ) ).

% real_arch_inverse
tff(fact_8690_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X)),X) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_8691_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).

% ln_inverse
tff(fact_8692_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X)))) ) ).

% plus_inverse_ge_2
tff(fact_8693_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_nat_real(power_power_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8694_real__le__x__sinh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% real_le_x_sinh
tff(fact_8695_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
          | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
      & ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
            | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
       => ( ( ( K = zero_zero_int )
           => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = L ) )
          & ( ( K != zero_zero_int )
           => ( ( ( L = zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = K ) )
              & ( ( L != zero_zero_int )
               => ( aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_8696_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,N2),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(N2,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_8697_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sinh_real,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_8698_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).

% sinh_real_less_iff
tff(fact_8699_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).

% sinh_real_pos_iff
tff(fact_8700_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).

% sinh_real_neg_iff
tff(fact_8701_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sinh_real_nonpos_iff
tff(fact_8702_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_8703_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_8704_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_8705_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% or_negative_int_iff
tff(fact_8706_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(N2,K),L)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_8707_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_concat_bit(N2,K),L)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).

% concat_bit_negative_iff
tff(fact_8708_cosh__real__nonzero,axiom,
    ! [X: real] : aa_real_real(cosh_real,X) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_8709_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).

% sinh_less_cosh_real
tff(fact_8710_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(N2,K),aa_int_int(bit_concat_bit(M,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2),aa_int_int(bit_concat_bit(N2,K),L)),R2) ).

% concat_bit_assoc
tff(fact_8711_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(X),Y))) ) ) ).

% OR_lower
tff(fact_8712_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se1409905431419307370or_int(K),L))) ) ).

% or_greater_eq
tff(fact_8713_cosh__real__pos,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_pos
tff(fact_8714_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_8715_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X))) ).

% cosh_real_nonneg
tff(fact_8716_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_8717_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_8718_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_8719_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_8720_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_8721_OR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ) ) ).

% OR_upper
tff(fact_8722_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8723_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8724_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real)) ) ) ).

% cot_less_zero
tff(fact_8725_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
                      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
                       => ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
                   => ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_8726_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_8727_Suc__0__mod__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N2),aa_nat_nat(suc,zero_zero_nat)))) ).

% Suc_0_mod_eq
tff(fact_8728_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_8729_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_8730_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_8731_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_8732_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_8733_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_8734_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_8735_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_8736_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_8737_log__eq__one,axiom,
    ! [A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_8738_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : adjust_div(product_Pair_int_int(Q4,R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q4),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_8739_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_8740_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_8741_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_8742_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_8743_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_8744_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_8745_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
          <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_8746_cot__npi,axiom,
    ! [N2: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_8747_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(power_power_real(A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_8748_or__Suc__0__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N2),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% or_Suc_0_eq
tff(fact_8749_Suc__0__or__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% Suc_0_or_eq
tff(fact_8750_or__nat__rec,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_8751_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).

% log_base_change
tff(fact_8752_less__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(power_power_real(B),N2)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).

% less_log_of_power
tff(fact_8753_log__of__power__eq,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(power_power_real(B),N2) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( aa_nat_real(semiri5074537144036343181t_real,N2) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_8754_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_8755_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_8756_le__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(power_power_real(B),N2)),M))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).

% le_log_of_power
tff(fact_8757_log__base__pow,axiom,
    ! [A: real,N2: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(power_power_real(A),N2)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_nat_real(semiri5074537144036343181t_real,N2)) ) ) ).

% log_base_pow
tff(fact_8758_log__nat__power,axiom,
    ! [X: real,B: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(power_power_real(X),N2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_nat_power
tff(fact_8759_log__inverse,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) ) ) ) ) ).

% log_inverse
tff(fact_8760_log__of__power__less,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),N2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N2))) ) ) ) ).

% log_of_power_less
tff(fact_8761_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
         => ( ( B != one_one_real )
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_8762_log__of__power__le,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(power_power_real(B),N2)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N2))) ) ) ) ).

% log_of_power_le
tff(fact_8763_less__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% less_log2_of_power
tff(fact_8764_le__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),M))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).

% le_log2_of_power
tff(fact_8765_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_vb(L)),Qr) ).

% divmod_step_nat_def
tff(fact_8766_log2__of__power__less,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N2))) ) ) ).

% log2_of_power_less
tff(fact_8767_or__nat__unfold,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N2) = N2 ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N2 = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N2) = M ) )
          & ( ( N2 != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_8768_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_vc(L),Qr) ).

% divmod_step_int_def
tff(fact_8769_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X))) ) ) ).

% cot_gt_zero
tff(fact_8770_log2__of__power__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N2))) ) ) ).

% log2_of_power_le
tff(fact_8771_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_8772_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_8773_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_8774_ceiling__log2__div2,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),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_8775_divmod__nat__if,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( ( N2 = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) )
       => ( divmod_nat(M,N2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
      & ( ~ ( ( N2 = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) )
       => ( divmod_nat(M,N2) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_vd),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2),N2)) ) ) ) ).

% divmod_nat_if
tff(fact_8776_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N2) )
        <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8777_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_8778_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_8779_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B))) = aa_int_int(divide_divide_int(one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_8780_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_8781_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_8782_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(divide_divide_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_8783_floor__eq,axiom,
    ! [N2: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N2)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N2)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N2 ) ) ) ).

% floor_eq
tff(fact_8784_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_gt
tff(fact_8785_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real))) ).

% real_of_int_floor_add_one_ge
tff(fact_8786_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_8787_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_8788_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_ve) ).

% prod_encode_def
tff(fact_8789_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_vf,Qr) ).

% Divides.adjust_div_def
tff(fact_8790_floor__eq2,axiom,
    ! [N2: int,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N2)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N2)),one_one_real)))
       => ( aa_real_int(archim6058952711729229775r_real,X) = N2 ) ) ) ).

% floor_eq2
tff(fact_8791_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(divide_divide_int(aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_8792_divmod__nat__def,axiom,
    ! [M: nat,N2: nat] : divmod_nat(M,N2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)) ).

% divmod_nat_def
tff(fact_8793_floor__log2__div2,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_8794_floor__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_8795_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8796_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),A0))
     => ( ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
       => ( ! [N: nat] :
              ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N)))
             => ( ! [X2: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4) = aa_nat7603221163087113075at_nat(nat_prod_decode,N) )
                   => pp(aa_nat_bool(P,Y4)) )
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N))) ) )
         => pp(aa_nat_bool(P,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_8797_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_vg(D))) ).

% int_ge_less_than_def
tff(fact_8798_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)))
    <=> ( X != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8799_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real))
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8800_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel_iff
tff(fact_8801_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8802_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_8803_powr__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X ) ) ).

% powr_one
tff(fact_8804_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).

% powr_le_cancel_iff
tff(fact_8805_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_8806_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_8807_powr__numeral,axiom,
    ! [X: real,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_num_real(numeral_numeral_real,N2)) = aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,N2)) ) ) ).

% powr_numeral
tff(fact_8808_powr__non__neg,axiom,
    ! [A: real,X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real)) ).

% powr_non_neg
tff(fact_8809_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_8810_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ).

% powr_ge_pzero
tff(fact_8811_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).

% powr_mono2
tff(fact_8812_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).

% powr_less_cancel
tff(fact_8813_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))) ) ) ).

% powr_less_mono
tff(fact_8814_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).

% powr_mono2'
tff(fact_8815_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).

% powr_less_mono2
tff(fact_8816_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ) ) ).

% gr_one_powr
tff(fact_8817_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_8818_powr__le1,axiom,
    ! [A: real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),one_one_real)) ) ) ) ).

% powr_le1
tff(fact_8819_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B))) ) ) ) ) ).

% powr_mono_both
tff(fact_8820_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8821_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_8822_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_8823_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ).

% inverse_powr
tff(fact_8824_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),B) ) ) ).

% log_base_powr
tff(fact_8825_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_powr
tff(fact_8826_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).

% log_powr
tff(fact_8827_powr__realpow,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,N2)) = aa_nat_real(power_power_real(X),N2) ) ) ).

% powr_realpow
tff(fact_8828_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_less_iff
tff(fact_8829_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_8830_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_less_iff
tff(fact_8831_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_8832_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_8833_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_8834_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).

% powr_le_iff
tff(fact_8835_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_8836_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).

% log_le_iff
tff(fact_8837_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_8838_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),A))) ) ) ).

% ln_powr_bound
tff(fact_8839_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),X))) ) ) ).

% ln_powr_bound2
tff(fact_8840_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8841_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8842_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8843_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
     => ( ( B != one_one_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8844_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_8845_powr__neg__numeral,axiom,
    ! [X: real,N2: num] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,N2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,N2))) ) ) ).

% powr_neg_numeral
tff(fact_8846_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = K )
        <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),X))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_8847_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_vh(D))) ).

% int_ge_less_than2_def
tff(fact_8848_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_vi,Y)) ).

% arctan_def
tff(fact_8849_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( aa_real_complex(complex2(X),Y) = aa_real_complex(real_V4546457046886955230omplex,Xa2) )
    <=> ( ( X = Xa2 )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_8850_complex__of__real__code,axiom,
    ! [X2: real] : aa_real_complex(real_V4546457046886955230omplex,X2) = aa_real_complex(complex2(X2),zero_zero_real) ).

% complex_of_real_code
tff(fact_8851_complex__of__real__def,axiom,
    ! [R2: real] : aa_real_complex(real_V4546457046886955230omplex,R2) = aa_real_complex(complex2(R2),zero_zero_real) ).

% complex_of_real_def
tff(fact_8852_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_vj) ) ) ).

% ln_neg_is_const
tff(fact_8853_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_vk) ).

% pi_half
tff(fact_8854_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_vk)) ).

% pi_def
tff(fact_8855_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N2))) = 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,N2),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N2),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_8856_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(power_power_real(X),aa_int_nat(nat2,I)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I)))) ) ) ) ) ).

% powr_int
tff(fact_8857_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_8858_nat__int,axiom,
    ! [N2: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,N2)) = N2 ).

% nat_int
tff(fact_8859_concat__bit__of__zero__2,axiom,
    ! [N2: nat,K: int] : aa_int_int(bit_concat_bit(N2,K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(N2),K) ).

% concat_bit_of_zero_2
tff(fact_8860_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_8861_nat__of__bool,axiom,
    ! [P: bool] : aa_int_nat(nat2,aa_bool_int(zero_n2684676970156552555ol_int,P)) = zero_n2687167440665602831ol_nat(P) ).

% nat_of_bool
tff(fact_8862_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_8863_nat__le__0,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int))
     => ( aa_int_nat(nat2,Z2) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_8864_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int)) ) ).

% nat_0_iff
tff(fact_8865_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ) ).

% zless_nat_conj
tff(fact_8866_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_8867_nat__zminus__int,axiom,
    ! [N2: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_8868_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = Z2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
       => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = zero_zero_int ) ) ) ).

% int_nat_eq
tff(fact_8869_take__bit__of__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ).

% take_bit_of_Suc_0
tff(fact_8870_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_8871_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_8872_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_8873_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_8874_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2)) ) ).

% zero_less_nat_eq
tff(fact_8875_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_8876_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_8877_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_8878_nat__abs__dvd__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,N2))) ) ).

% nat_abs_dvd_iff
tff(fact_8879_dvd__nat__abs__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N2),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,N2)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_8880_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).

% nat_ceiling_le_eq
tff(fact_8881_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z2)) ) ).

% one_less_nat_eq
tff(fact_8882_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_8883_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_8884_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_8885_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2)),aa_int_nat(nat2,A)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2)),A)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_8886_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N2))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_8887_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),Q4))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_8888_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_8889_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_8890_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N2),K)) ) ) ).

% take_bit_nat_eq
tff(fact_8891_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_8892_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(N2),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_8893_nat__numeral__as__int,axiom,
    ! [X2: num] : aa_num_nat(numeral_numeral_nat,X2) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X2)) ).

% nat_numeral_as_int
tff(fact_8894_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N2),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_8895_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(N2),K))) ).

% take_bit_nonnegative
tff(fact_8896_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_8897_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N2),K)),zero_zero_int)) ).

% not_take_bit_negative
tff(fact_8898_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).

% nat_mono
tff(fact_8899_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_8900_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
       => ( ( aa_int_nat(nat2,Z2) = aa_int_nat(nat2,Z5) )
        <=> ( Z2 = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_8901_all__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ! [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ! [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
         => pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% all_nat
tff(fact_8902_ex__nat,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_13: nat] : pp(aa_nat_bool(P,X_13))
    <=> ? [X3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
          & pp(aa_nat_bool(P,aa_int_nat(nat2,X3))) ) ) ).

% ex_nat
tff(fact_8903_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_8904_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_8905_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ) ).

% nat_mono_iff
tff(fact_8906_take__bit__decr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N2),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N2),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_8907_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z2)) ) ).

% zless_nat_eq_int_zless
tff(fact_8908_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N2))) ) ).

% nat_le_iff
tff(fact_8909_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z2 )
    <=> ( ( M = aa_int_nat(nat2,Z2) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) ) ) ).

% int_eq_iff
tff(fact_8910_nat__0__le,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = Z2 ) ) ).

% nat_0_le
tff(fact_8911_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_8912_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_8913_int__minus,axiom,
    ! [N2: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,N2)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_8914_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z2: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W2),Z2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z2))) ).

% nat_abs_mult_distrib
tff(fact_8915_nat__plus__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_8916_nat__times__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_8917_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))))) ).

% real_nat_ceiling_ge
tff(fact_8918_nat__minus__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_8919_nat__div__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),Xa) = aa_int_nat(nat2,aa_int_int(divide_divide_int(aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_8920_nat__mod__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),Xa) = aa_int_nat(nat2,aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_8921_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero_int )
       => ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
      & ( ( I != zero_zero_int )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
          & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
           => ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).

% zsgn_def
tff(fact_8922_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),Z2)) ) ) ).

% nat_less_eq_zless
tff(fact_8923_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W2))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W2),Z2)) ) ) ).

% nat_le_eq_zle
tff(fact_8924_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
tff(fact_8925_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( W2 = aa_nat_int(semiri1314217659103216013at_int,M) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
tff(fact_8926_split__nat,axiom,
    ! [P: fun_nat_bool,I: int] :
      ( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
    <=> ( ! [N4: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => pp(aa_nat_bool(P,N4)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
         => pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).

% split_nat
tff(fact_8927_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N2)),K)) ) ) ).

% le_nat_iff
tff(fact_8928_nat__add__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_8929_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_8930_nat__mult__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_8931_Suc__as__int,axiom,
    ! [X2: nat] : aa_nat_nat(suc,X2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),one_one_int)) ).

% Suc_as_int
tff(fact_8932_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_8933_nat__diff__distrib,axiom,
    ! [Z5: int,Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z5),Z2))
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_8934_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ).

% nat_abs_triangle_ineq
tff(fact_8935_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_8936_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( aa_int_nat(nat2,aa_int_int(divide_divide_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_8937_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
     => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_8938_nat__power__eq,axiom,
    ! [Z2: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z2),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z2)),N2) ) ) ).

% nat_power_eq
tff(fact_8939_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_8940_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( aa_int_nat(nat2,aa_int_int(modulo_modulo_int(X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_8941_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_8942_floor__eq3,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N2 ) ) ) ).

% floor_eq3
tff(fact_8943_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).

% le_nat_floor
tff(fact_8944_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_8945_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(divide_divide_int(K),L) = zero_zero_int ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_8946_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero_int )
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) )
      & ( ( L != zero_zero_int )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_8947_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_8948_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% take_bit_nat_less_exp
tff(fact_8949_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M) = M )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_8950_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_8951_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N2),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ).

% take_bit_int_less_exp
tff(fact_8952_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z2)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_8953_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W2)),M))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).

% nat_less_iff
tff(fact_8954_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z2),zero_zero_int))
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z2))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_8955_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
       => ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).

% nat_abs_int_diff
tff(fact_8956_floor__eq4,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N2 ) ) ) ).

% floor_eq4
tff(fact_8957_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_8958_diff__nat__eq__if,axiom,
    ! [Z5: int,Z2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) = aa_int_nat(nat2,Z2) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z5)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z5))) ) ) ) ).

% diff_nat_eq_if
tff(fact_8959_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),M)),M))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_8960_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_8961_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_8962_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N2),K)),K))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_8963_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_int_nat(nat2,Z2)),M))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z2),aa_nat_int(semiri1314217659103216013at_int,M))) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z2))
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
tff(fact_8964_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L)))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L)),R2) )
         => eucl_rel_int(K,L,product_Pair_int_int(Q4,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8965_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N2),K) = K )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_8966_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N2),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_8967_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L4: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q6,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q6: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q6,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4)))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q6),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8968_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q2: int] :
              ( ( A32 = product_Pair_int_int(Q2,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22) ) ) )
         => ~ ! [R3: int,Q2: int] :
                ( ( A32 = product_Pair_int_int(Q2,R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22)))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8969_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(divide_divide_int(K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(divide_divide_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_8970_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N2),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))) ) ) ).

% take_bit_int_less_eq
tff(fact_8971_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))),aa_int_int(bit_se2923211474154528505it_int(N2),K))) ) ).

% take_bit_int_greater_eq
tff(fact_8972_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).

% even_nat_iff
tff(fact_8973_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N2: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N2),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_8974_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))
       => ( aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_8975_powr__real__of__int,axiom,
    ! [X: real,N2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N2)) = aa_nat_real(power_power_real(X),aa_int_nat(nat2,N2)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N2))
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N2)) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N2)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_8976_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_8977_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N2))) = zero_zero_int ) )
      & ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
            | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N2))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)) ) )
          & ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
           => ( aa_int_int(divide_divide_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N2))) = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2)),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N2),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_8978_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_8979_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero_int )
          | ( L = zero_zero_int ) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = zero_zero_int ) )
      & ( ~ ( ( K = zero_zero_int )
            | ( L = zero_zero_int ) )
       => ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = L ) )
          & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = K ) )
              & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
               => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_8980_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_8981_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_8982_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N2) )
    <=> ( aa_num_nat(pred_numeral,K) = N2 ) ) ).

% eq_numeral_Suc
tff(fact_8983_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( aa_nat_nat(suc,N2) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N2 = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_8984_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_8985_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% and_negative_int_iff
tff(fact_8986_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X)))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).

% zero_le_sgn_iff
tff(fact_8987_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).

% sgn_le_0_iff
tff(fact_8988_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_8989_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_num_nat(pred_numeral,K))) ) ).

% less_Suc_numeral
tff(fact_8990_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N2)) ) ).

% less_numeral_Suc
tff(fact_8991_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_8992_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_num_nat(pred_numeral,K))) ) ).

% le_Suc_numeral
tff(fact_8993_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N2)) ) ).

% le_numeral_Suc
tff(fact_8994_diff__Suc__numeral,axiom,
    ! [N2: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_8995_diff__numeral__Suc,axiom,
    ! [K: num,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N2) ).

% diff_numeral_Suc
tff(fact_8996_max__Suc__numeral,axiom,
    ! [N2: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N2),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_8997_max__numeral__Suc,axiom,
    ! [K: num,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N2)) ).

% max_numeral_Suc
tff(fact_8998_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_8999_and__minus__numerals_I5_J,axiom,
    ! [N2: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9000_and__minus__numerals_I1_J,axiom,
    ! [N2: 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,N2)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9001_num__induct,axiom,
    ! [P: fun_num_bool,X: num] :
      ( pp(aa_num_bool(P,one))
     => ( ! [X4: num] :
            ( pp(aa_num_bool(P,X4))
           => pp(aa_num_bool(P,aa_num_num(inc,X4))) )
       => pp(aa_num_bool(P,X)) ) ) ).

% num_induct
tff(fact_9002_add__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(plus_plus_num(X),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(plus_plus_num(X),Y)) ).

% add_inc
tff(fact_9003_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(X),Y))) ) ).

% AND_lower
tff(fact_9004_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),X)) ) ).

% AND_upper1
tff(fact_9005_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Y)) ) ).

% AND_upper2
tff(fact_9006_AND__upper1_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z2)) ) ) ).

% AND_upper1'
tff(fact_9007_AND__upper2_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z2)) ) ) ).

% AND_upper2'
tff(fact_9008_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_9009_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9010_inc_Osimps_I3_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit1,X)) = aa_num_num(bit0,aa_num_num(inc,X)) ).

% inc.simps(3)
tff(fact_9011_inc_Osimps_I2_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_9012_add__One,axiom,
    ! [X: num] : aa_num_num(plus_plus_num(X),one) = aa_num_num(inc,X) ).

% add_One
tff(fact_9013_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K)) ) ).

% and_less_eq
tff(fact_9014_AND__upper1_H_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Z2)) ) ) ).

% AND_upper1''
tff(fact_9015_AND__upper2_H_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(X),Y)),Z2)) ) ) ).

% AND_upper2''
tff(fact_9016_inc__BitM__eq,axiom,
    ! [N2: num] : aa_num_num(inc,aa_num_num(bitM,N2)) = aa_num_num(bit0,N2) ).

% inc_BitM_eq
tff(fact_9017_BitM__inc__eq,axiom,
    ! [N2: num] : aa_num_num(bitM,aa_num_num(inc,N2)) = aa_num_num(bit1,N2) ).

% BitM_inc_eq
tff(fact_9018_mult__inc,axiom,
    ! [X: num,Y: num] : aa_num_num(times_times_num(X),aa_num_num(inc,Y)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(X),Y)),X) ).

% mult_inc
tff(fact_9019_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_9020_sgn__real__def,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( aa_real_real(sgn_sgn_real,A) = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) )
          & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
           => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).

% sgn_real_def
tff(fact_9021_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,A)),N2)) = X )
     => ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,B)),N2)) )
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9022_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
          & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
      & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
       => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_9023_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_9024_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N2: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N2),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N2))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N2)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9025_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ).

% mask_nat_positive_iff
tff(fact_9026_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9027_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_9028_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(N2),K)))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9029_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N2),K)),zero_zero_int))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) ).

% signed_take_bit_negative_iff
tff(fact_9030_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9031_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_9032_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2)))),aa_nat_nat(suc,N2)))
    <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),N2)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9033_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2)))),aa_nat_nat(suc,N2)))
    <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W2)),N2)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9034_Suc__0__and__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9035_and__Suc__0__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N2),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9036_less__eq__mask,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(bit_se2002935070580805687sk_nat,N2))) ).

% less_eq_mask
tff(fact_9037_mask__nonnegative__int,axiom,
    ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N2))) ).

% mask_nonnegative_int
tff(fact_9038_not__mask__negative__int,axiom,
    ! [N2: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N2)),zero_zero_int)) ).

% not_mask_negative_int
tff(fact_9039_less__mask,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_nat_nat(bit_se2002935070580805687sk_nat,N2))) ) ).

% less_mask
tff(fact_9040_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
     => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9041_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(M,K),L)),N2))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),M))
          & pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
          & pp(aa_nat_bool(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9042_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2)))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N2),set_or1266510415728281911st_int(M,N2)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_9043_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_9044_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N2),K) = aa_nat_int(bit_se2000444600071755411sk_int,N2) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(N2),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9045_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
             => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),M2))
              <=> pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
             => ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))
              <=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ) ) ).

% int_bit_bound
tff(fact_9046_Suc__mask__eq__exp,axiom,
    ! [N2: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2) ).

% Suc_mask_eq_exp
tff(fact_9047_mask__nat__less__exp,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ).

% mask_nat_less_exp
tff(fact_9048_mask__nat__def,axiom,
    ! [N2: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N2) = 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))),N2)),one_one_nat) ).

% mask_nat_def
tff(fact_9049_mask__half__int,axiom,
    ! [N2: nat] : aa_int_int(divide_divide_int(aa_nat_int(bit_se2000444600071755411sk_int,N2)),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,N2),one_one_nat)) ).

% mask_half_int
tff(fact_9050_and__nat__unfold,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ( M = zero_zero_nat )
          | ( N2 = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N2) = zero_zero_nat ) )
      & ( ~ ( ( M = zero_zero_nat )
            | ( N2 = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_9051_and__nat__rec,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9052_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N2)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N2)))),aa_int_int(bit_se2923211474154528505it_int(N2),K)) ).

% take_bit_Suc_from_most
tff(fact_9053_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K,L))
     => ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
            & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
        & ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
              & pp(aa_set_int_bool(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
         => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_9054_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(X),Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa2))
       => ~ ( ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                  & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
              & ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(divide_divide_int(X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(X,Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_9055_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9056_rat__inverse__code,axiom,
    ! [P3: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P3)) = produc4245557441103728435nt_int(aTP_Lamm_vl,quotient_of(P3)) ).

% rat_inverse_code
tff(fact_9057_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_9058_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_9059_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_9060_set__encode__insert,axiom,
    ! [A2: set_nat,N2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ~ pp(aa_set_nat_bool(member_nat(N2),A2))
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N2,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_9061_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N2))) ).

% not_bit_Suc_0_Suc
tff(fact_9062_bit__Suc__0__iff,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N2))
    <=> ( N2 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9063_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_9064_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_9065_not__bit__Suc__0__numeral,axiom,
    ! [N2: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N2))) ).

% not_bit_Suc_0_numeral
tff(fact_9066_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2)) = insert_nat(aa_nat_nat(suc,N2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2)) ).

% atLeast0_atMost_Suc
tff(fact_9067_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_9068_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N2)))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N2)) = insert_nat(aa_nat_nat(suc,N2),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_9069_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N2)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),N2) ) ) ).

% atLeastAtMost_insertL
tff(fact_9070_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_9071_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9072_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q4: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% quotient_of_denom_pos
tff(fact_9073_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
        & pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) ) ).

% bit_nat_iff
tff(fact_9074_rat__less__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P3),Q4))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_vn(Q4)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_9075_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9076_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_9077_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N2) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N2),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9078_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9079_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z2: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(N2),aa_nat_set_nat(nat_set_decode,Z2)))
     => ( 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))),N2)),Z2)) = insert_nat(N2,aa_nat_set_nat(nat_set_decode,Z2)) ) ) ).

% set_decode_plus_power_2
tff(fact_9080_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(A0,A1))
     => ( ! [K2: int,L3: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(K2,L3))
           => ( ( ~ ( pp(aa_set_int_bool(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
                    & pp(aa_set_int_bool(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(divide_divide_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K2),L3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_9081_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ).

% arccos_le_pi2
tff(fact_9082_normalize__negative,axiom,
    ! [Q4: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q4),zero_zero_int))
     => ( normalize(product_Pair_int_int(P3,Q4)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q4))) ) ) ).

% normalize_negative
tff(fact_9083_xor__Suc__0__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N2),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,N2),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) ).

% xor_Suc_0_eq
tff(fact_9084_Suc__0__xor__eq,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) ).

% Suc_0_xor_eq
tff(fact_9085_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9086_normalize__denom__zero,axiom,
    ! [P3: int] : normalize(product_Pair_int_int(P3,zero_zero_int)) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% normalize_denom_zero
tff(fact_9087_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_9088_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_9089_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_9090_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_9091_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_9092_sgn__rat__def,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) )
          & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
           => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_9093_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9094_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
     => ~ ! [S2: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S2))
           => ! [T2: rat] :
                ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T2))
               => ( R2 != aa_rat_rat(plus_plus_rat(S2),T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9095_abs__rat__def,axiom,
    ! [A: rat] :
      ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) )
      & ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
       => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ) ).

% abs_rat_def
tff(fact_9096_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_9097_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_9098_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
        <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_9099_arccos__cos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_9100_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q4: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q4) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4)) ) ).

% normalize_denom_pos
tff(fact_9101_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P3: int,R2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P3,Q4)) = normalize(product_Pair_int_int(R2,S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P3),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9102_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_9103_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_9104_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9105_arccos__cos2,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = aa_real_real(uminus_uminus_real,X) ) ) ) ).

% arccos_cos2
tff(fact_9106_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_vo,Y)) ).

% arccos_def
tff(fact_9107_arccos,axiom,
    ! [Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi))
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_9108_xor__nat__unfold,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N2) = N2 ) )
      & ( ( M != zero_zero_nat )
       => ( ( ( N2 = zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N2) = M ) )
          & ( ( N2 != zero_zero_nat )
           => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_9109_xor__nat__rec,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9110_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_bool(Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_9111_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K),L)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9112_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9113_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int))
    <=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).

% xor_negative_int_iff
tff(fact_9114_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N2),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% push_bit_negative_int_iff
tff(fact_9115_concat__bit__of__zero__1,axiom,
    ! [N2: nat,L: int] : aa_int_int(bit_concat_bit(N2,zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(N2),L) ).

% concat_bit_of_zero_1
tff(fact_9116_push__bit__of__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N2),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))),N2) ).

% push_bit_of_Suc_0
tff(fact_9117_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(X),Y))) ) ) ).

% XOR_lower
tff(fact_9118_set__bit__nat__def,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9119_flip__bit__nat__def,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N2) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9120_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(M),K)),N2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
        & pp(aa_nat_bool(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_9121_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q4)),N2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9122_XOR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N2))) ) ) ) ).

% XOR_upper
tff(fact_9123_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
      & ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
          & ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
           => ( ( ( K = zero_zero_int )
               => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = L ) )
              & ( ( K != zero_zero_int )
               => ( ( ( L = zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = K ) )
                  & ( ( L != zero_zero_int )
                   => ( aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(divide_divide_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_9124_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N2)) = 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,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9125_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),M6))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M9),N4))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M6)),aa_nat_real(X5,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_9126_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9127_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U))) ).

% finite_atLeastLessThan
tff(fact_9128_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% not_nonnegative_int_iff
tff(fact_9129_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% not_negative_int_iff
tff(fact_9130_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9131_all__nat__less__eq,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N2))
         => pp(aa_nat_bool(P,M6)) )
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2)))
         => pp(aa_nat_bool(P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_9132_ex__nat__less__eq,axiom,
    ! [N2: nat,P: fun_nat_bool] :
      ( ? [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N2))
          & pp(aa_nat_bool(P,M6)) )
    <=> ? [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2)))
          & pp(aa_nat_bool(P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_9133_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_9134_lessThan__atLeast0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N2) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2) ).

% lessThan_atLeast0
tff(fact_9135_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9136_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_9137_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_9138_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N2)) = insert_nat(N2,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2)) ).

% atLeast0_lessThan_Suc
tff(fact_9139_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2)))
     => pp(aa_set_nat_bool(finite_finite_nat,N5)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9140_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N2)) = insert_nat(N2,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N2)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_9141_prod__Suc__fact,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2)) = aa_nat_nat(semiri1408675320244567234ct_nat,N2) ).

% prod_Suc_fact
tff(fact_9142_prod__Suc__Suc__fact,axiom,
    ! [N2: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N2)) = aa_nat_nat(semiri1408675320244567234ct_nat,N2) ).

% prod_Suc_Suc_fact
tff(fact_9143_and__not__numerals_I3_J,axiom,
    ! [N2: 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,N2)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9144_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9145_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
       => ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_9146_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N2) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N2),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9147_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2))) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vp(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2)))),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),N2))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9148_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(L,U))) ).

% finite_atLeastLessThan_int
tff(fact_9149_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa_set_int_bool(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_9150_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_9151_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_9152_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_9153_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_vq),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_9154_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_vr),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_9155_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N2))
    <=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N2)) ) ).

% less_eq_nat.simps(2)
tff(fact_9156_max__Suc2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N2),aa_nat_fun_nat_nat(aTP_Lamm_vs,N2)),M) ).

% max_Suc2
tff(fact_9157_max__Suc1,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N2)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N2),aa_nat_fun_nat_nat(aTP_Lamm_vt,N2)),M) ).

% max_Suc1
tff(fact_9158_diff__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_au),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)) ).

% diff_Suc
tff(fact_9159_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_au),Nat) ).

% pred_def
tff(fact_9160_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_vu(X)) ).

% floor_rat_def
tff(fact_9161_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aTP_Lamm_vv(X)) ).

% floor_real_def
tff(fact_9162_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),YS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_max_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_9163_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_9164_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_9165_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9166_snd__divmod__nat,axiom,
    ! [M: nat,N2: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N2) ).

% snd_divmod_nat
tff(fact_9167_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat),fun_min_weak) ).

% wmin_emptyI
tff(fact_9168_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X5))
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X5),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_9169_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_9170_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),Y5))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y5),fun_max_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X5),Y5),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_9171_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9172_fst__divmod__nat,axiom,
    ! [M: nat,N2: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(M,N2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) ).

% fst_divmod_nat
tff(fact_9173_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2)))) ).

% quotient_of_denom_pos'
tff(fact_9174_smax__emptyI,axiom,
    ! [Y5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,Y5))
     => ( ( Y5 != bot_bo2099793752762293965at_nat )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y5),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_9175_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,bot_bo2099793752762293965at_nat),fun_min_strict) ) ).

% smin_emptyI
tff(fact_9176_minus__one__mod__numeral,axiom,
    ! [N2: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N2)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N2)),product_snd_int_int(unique5052692396658037445od_int(one,N2))) ).

% minus_one_mod_numeral
tff(fact_9177_one__mod__minus__numeral,axiom,
    ! [N2: num] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N2)),product_snd_int_int(unique5052692396658037445od_int(one,N2)))) ).

% one_mod_minus_numeral
tff(fact_9178_numeral__mod__minus__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N2)),product_snd_int_int(unique5052692396658037445od_int(M,N2)))) ).

% numeral_mod_minus_numeral
tff(fact_9179_minus__numeral__mod__numeral,axiom,
    ! [M: num,N2: num] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,N2)),product_snd_int_int(unique5052692396658037445od_int(M,N2))) ).

% minus_numeral_mod_numeral
tff(fact_9180_Divides_Oadjust__mod__def,axiom,
    ! [R2: int,L: int] :
      ( ( ( R2 = zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R2) = zero_zero_int ) )
      & ( ( R2 != zero_zero_int )
       => ( aa_int_int(adjust_mod(L),R2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_9181_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_9182_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(one_one_int,zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_9183_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_9184_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = product_Pair_int_int(one_one_int,zero_zero_int) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% bezw.pelims
tff(fact_9185_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( normalize(P3) = product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))),aa_int_int(divide_divide_int(product_snd_int_int(P3)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P3)))
       => ( ( ( product_snd_int_int(P3) = zero_zero_int )
           => ( normalize(P3) = product_Pair_int_int(zero_zero_int,one_one_int) ) )
          & ( ( product_snd_int_int(P3) != zero_zero_int )
           => ( normalize(P3) = product_Pair_int_int(aa_int_int(divide_divide_int(product_fst_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3)))),aa_int_int(divide_divide_int(product_snd_int_int(P3)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P3)),product_snd_int_int(P3))))) ) ) ) ) ) ).

% normalize_def
tff(fact_9186_gcd__1__int,axiom,
    ! [M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9187_gcd__neg1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_neg1_int
tff(fact_9188_gcd__neg2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_neg2_int
tff(fact_9189_gcd__abs2__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(abs_abs_int,Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_abs2_int
tff(fact_9190_gcd__abs1__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(abs_abs_int,X)),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% gcd_abs1_int
tff(fact_9191_abs__gcd__int,axiom,
    ! [X: int,Y: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% abs_gcd_int
tff(fact_9192_gcd__pos__int,axiom,
    ! [M: int,N2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N2)))
    <=> ( ( M != zero_zero_int )
        | ( N2 != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9193_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N2: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_num_int(numeral_numeral_int,N2)) ).

% gcd_neg_numeral_2_int
tff(fact_9194_gcd__neg__numeral__1__int,axiom,
    ! [N2: num,X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))),X) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N2)),X) ).

% gcd_neg_numeral_1_int
tff(fact_9195_gcd__0__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = aa_int_int(abs_abs_int,X) ).

% gcd_0_int
tff(fact_9196_gcd__0__left__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),X) = aa_int_int(abs_abs_int,X) ).

% gcd_0_left_int
tff(fact_9197_gcd__proj2__if__dvd__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(abs_abs_int,Y) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_9198_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(abs_abs_int,X) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_9199_gcd__red__int,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(modulo_modulo_int(X),Y)) ).

% gcd_red_int
tff(fact_9200_gcd__idem__int,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),X) = aa_int_int(abs_abs_int,X) ).

% gcd_idem_int
tff(fact_9201_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ).

% gcd_ge_0_int
tff(fact_9202_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) ).

% bezout_int
tff(fact_9203_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N2)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N2)) ).

% gcd_mult_distrib_int
tff(fact_9204_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A)) ) ).

% gcd_le1_int
tff(fact_9205_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B)) ) ).

% gcd_le2_int
tff(fact_9206_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_bool] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y)))) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y))) ) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y)))) ) )
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y))) ) ) ) ) ).

% gcd_cases_int
tff(fact_9207_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),D))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),A))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
        & ! [E3: int] :
            ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),A))
              & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),B)) )
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,E3),D)) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9208_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(modulo_modulo_int(X),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9209_gcd__code__int,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),L) = aa_int_int(abs_abs_int,aa_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,L),zero_zero_int),K),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(modulo_modulo_int(aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9210_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( aa_nat7603221163087113075at_nat(nat_prod_decode_aux(X),Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
               => ( Y = aa_nat7603221163087113075at_nat(nat_prod_decode_aux(aa_nat_nat(suc,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9211_root__powr__inverse,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N2),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N2))) ) ) ) ).

% root_powr_inverse
tff(fact_9212_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9213_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ).

% gcd_nat.left_neutral
tff(fact_9214_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9215_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ).

% gcd_nat.right_neutral
tff(fact_9216_gcd__0__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).

% gcd_0_nat
tff(fact_9217_gcd__0__left__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).

% gcd_0_left_nat
tff(fact_9218_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9219_gcd__nat_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb1
tff(fact_9220_gcd__nat_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb2
tff(fact_9221_gcd__nat_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% gcd_nat.bounded_iff
tff(fact_9222_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_9223_gcd__proj2__if__dvd__nat,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_9224_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9225_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2)))
    <=> ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9226_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(N2),K)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9227_drop__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N2),K)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).

% drop_bit_negative_int_iff
tff(fact_9228_real__root__zero,axiom,
    ! [N2: nat] : aa_real_real(aa_nat_fun_real_real(root,N2),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9229_gcd__int__int__eq,axiom,
    ! [M: nat,N2: nat] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2)) ).

% gcd_int_int_eq
tff(fact_9230_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_9231_real__root__eq__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N2),X) = aa_real_real(aa_nat_fun_real_real(root,N2),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9232_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_9233_drop__bit__of__Suc__0,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N2),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N2),zero_zero_nat)) ).

% drop_bit_of_Suc_0
tff(fact_9234_drop__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(N2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9235_real__root__eq__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N2),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9236_real__root__less__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).

% real_root_less_iff
tff(fact_9237_real__root__le__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).

% real_root_le_iff
tff(fact_9238_real__root__one,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_nat_fun_real_real(root,N2),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9239_real__root__eq__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N2),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9240_gcd__nat__abs__right__eq,axiom,
    ! [N2: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,N2),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_nat_int(semiri1314217659103216013at_int,N2)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_9241_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),aa_nat_int(semiri1314217659103216013at_int,N2))) ).

% gcd_nat_abs_left_eq
tff(fact_9242_real__root__lt__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).

% real_root_lt_0_iff
tff(fact_9243_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_9244_real__root__le__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),zero_zero_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).

% real_root_le_0_iff
tff(fact_9245_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_9246_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_9247_real__root__lt__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).

% real_root_lt_1_iff
tff(fact_9248_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N2),Y)))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_9249_real__root__le__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),one_one_real))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).

% real_root_le_1_iff
tff(fact_9250_drop__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N2)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(N2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9251_real__root__pow__pos2,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),N2) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_9252_gcd__nat_Omono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),D))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,C),D))) ) ) ).

% gcd_nat.mono
tff(fact_9253_gcd__nat_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.orderE
tff(fact_9254_gcd__nat_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).

% gcd_nat.orderI
tff(fact_9255_gcd__nat_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb3
tff(fact_9256_gcd__nat_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
        & ( B != A ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb4
tff(fact_9257_gcd__nat_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).

% gcd_nat.boundedE
tff(fact_9258_gcd__nat_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C))) ) ) ).

% gcd_nat.boundedI
tff(fact_9259_gcd__nat_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_nat.order_iff
tff(fact_9260_gcd__nat_Ocobounded1,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ).

% gcd_nat.cobounded1
tff(fact_9261_gcd__nat_Ocobounded2,axiom,
    ! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ).

% gcd_nat.cobounded2
tff(fact_9262_gcd__nat_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = A ) ) ).

% gcd_nat.absorb_iff1
tff(fact_9263_gcd__nat_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = B ) ) ).

% gcd_nat.absorb_iff2
tff(fact_9264_gcd__nat_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)) ) ).

% gcd_nat.coboundedI1
tff(fact_9265_gcd__nat_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C)) ) ).

% gcd_nat.coboundedI2
tff(fact_9266_gcd__nat_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C)))
        & ( A != aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,B),C) ) )
     => ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
            & ( A != B ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
              & ( A != C ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_9267_gcd__nat_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
        & ( A != B ) )
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
        & ( A != B ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_9268_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
        & ( A != C ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_9269_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
        & ( B != C ) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),C))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) != C ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_9270_gcd__unique__nat,axiom,
    ! [D: nat,A: nat,B: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
        & ! [E3: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E3),A))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E3),B)) )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,E3),D)) ) )
    <=> ( D = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ).

% gcd_unique_nat
tff(fact_9271_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N2)) ).

% gcd_mult_distrib_nat
tff(fact_9272_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ).

% gcd_red_nat
tff(fact_9273_real__root__pos__pos__le,axiom,
    ! [X: real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N2),X))) ) ).

% real_root_pos_pos_le
tff(fact_9274_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ) ).

% gcd_le2_nat
tff(fact_9275_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ) ).

% gcd_le1_nat
tff(fact_9276_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N2)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) ) ) ).

% gcd_diff1_nat
tff(fact_9277_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),M)),N2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) ) ) ).

% gcd_diff2_nat
tff(fact_9278_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_9279_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
      & ( ( Y != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_9280_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9281_real__root__less__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N2),Y))) ) ) ).

% real_root_less_mono
tff(fact_9282_real__root__le__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N2),Y))) ) ) ).

% real_root_le_mono
tff(fact_9283_real__root__power,axiom,
    ! [N2: nat,X: real,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_nat_fun_real_real(root,N2),aa_nat_real(power_power_real(X),K)) = aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),K) ) ) ).

% real_root_power
tff(fact_9284_real__root__abs,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_nat_fun_real_real(root,N2),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)) ) ) ).

% real_root_abs
tff(fact_9285_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_9286_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9287_sgn__root,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_9288_real__root__gt__zero,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N2),X))) ) ) ).

% real_root_gt_zero
tff(fact_9289_real__root__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N5),X)),aa_real_real(aa_nat_fun_real_real(root,N2),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9290_root__abs__power,axiom,
    ! [N2: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N2),aa_nat_real(power_power_real(Y),N2))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9291_gcd__int__def,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,X))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y)))) ).

% gcd_int_def
tff(fact_9292_real__root__pos__pos,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N2),X))) ) ) ).

% real_root_pos_pos
tff(fact_9293_real__root__strict__increasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9294_real__root__decreasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N5),X)),aa_real_real(aa_nat_fun_real_real(root,N2),X))) ) ) ) ).

% real_root_decreasing
tff(fact_9295_real__root__pow__pos,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => ( aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),N2) = X ) ) ) ).

% real_root_pow_pos
tff(fact_9296_real__root__pos__unique,axiom,
    ! [N2: nat,Y: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
       => ( ( aa_nat_real(power_power_real(Y),N2) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,N2),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9297_real__root__power__cancel,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
       => ( aa_real_real(aa_nat_fun_real_real(root,N2),aa_nat_real(power_power_real(X),N2)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_9298_real__root__increasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),N5))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,N5),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_9299_root__sgn__power,axiom,
    ! [N2: nat,Y: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_nat_fun_real_real(root,N2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y)),N2))) = Y ) ) ).

% root_sgn_power
tff(fact_9300_sgn__power__root,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N2),X))),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N2),X))),N2)) = X ) ) ).

% sgn_power_root
tff(fact_9301_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(X,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9302_ln__root,axiom,
    ! [N2: nat,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N2),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,N2)) ) ) ) ).

% ln_root
tff(fact_9303_log__root,axiom,
    ! [N2: nat,A: real,B: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,N2),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,N2)) ) ) ) ).

% log_root
tff(fact_9304_log__base__root,axiom,
    ! [N2: nat,B: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,N2),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).

% log_base_root
tff(fact_9305_split__root,axiom,
    ! [P: fun_real_bool,N2: nat,X: real] :
      ( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N2),X)))
    <=> ( ( ( N2 = zero_zero_nat )
         => pp(aa_real_bool(P,zero_zero_real)) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
         => ! [Y6: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y6)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Y6)),N2)) = X )
             => pp(aa_real_bool(P,Y6)) ) ) ) ) ).

% split_root
tff(fact_9306_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).

% gcd_nat.pelims
tff(fact_9307_gcd__idem__nat,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),X) = X ).

% gcd_idem_nat
tff(fact_9308_gcd__nat_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.right_idem
tff(fact_9309_gcd__nat_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ).

% gcd_nat.left_idem
tff(fact_9310_gcd__nat_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),A) = A ).

% gcd_nat.idem
tff(fact_9311_Suc__funpow,axiom,
    ! [N2: nat] : compow_fun_nat_nat(N2,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N2) ).

% Suc_funpow
tff(fact_9312_sub__BitM__One__eq,axiom,
    ! [N2: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,N2)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(N2),one)) ).

% sub_BitM_One_eq
tff(fact_9313_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_am,aTP_Lamm_al) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9314_card__lessThan,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_9315_card__Collect__less__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,N2))) = N2 ).

% card_Collect_less_nat
tff(fact_9316_card__atMost,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_9317_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_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_9318_card__Collect__le__nat,axiom,
    ! [N2: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_am,N2))) = aa_nat_nat(suc,N2) ).

% card_Collect_le_nat
tff(fact_9319_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9320_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_9321_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_9322_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_vw(M7),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_vx(M7),I))) ) ) ).

% card_less_Suc2
tff(fact_9323_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_vw(M7),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_vx(M7),I))) ) ) ).

% card_less_Suc
tff(fact_9324_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_vx(M7),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9325_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_9326_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2)))))
     => ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9327_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N2: nat] :
      ( pp(aa_set_nat_bool(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N5)),N2)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9328_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_au),S3))) ).

% card_sum_le_nat_sum
tff(fact_9329_card__nth__roots,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_vy(C),N2))) = N2 ) ) ) ).

% card_nth_roots
tff(fact_9330_card__roots__unity__eq,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( finite_card_complex(collect_complex(aTP_Lamm_bq(N2))) = N2 ) ) ).

% card_roots_unity_eq
tff(fact_9331_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_vz) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9332_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_9333_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_9334_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ? [R3: fun_nat_nat] :
          ( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S3))))
          & ! [N6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N6),aa_set_nat_nat(finite_card_nat,S3)))
             => pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N6)),S3)) ) ) ) ).

% finite_enumerate
tff(fact_9335_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_9336_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M7,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9337_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or5832277885323065728an_int(L,U))) ).

% finite_greaterThanLessThan_int
tff(fact_9338_Gcd__dvd__nat,axiom,
    ! [A: nat,A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(A),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(gcd_Gcd_nat,A2)),A)) ) ).

% Gcd_dvd_nat
tff(fact_9339_Gcd__greatest__nat,axiom,
    ! [A2: set_nat,A: nat] :
      ( ! [B3: nat] :
          ( pp(aa_set_nat_bool(member_nat(B3),A2))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B3)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_set_nat_nat(gcd_Gcd_nat,A2))) ) ).

% Gcd_greatest_nat
tff(fact_9340_Gcd__nat__eq__one,axiom,
    ! [N5: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(one_one_nat),N5))
     => ( aa_set_nat_nat(gcd_Gcd_nat,N5) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9341_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_set_nat_bool(member_nat(A3),A2))
         => ( pp(aa_set_nat_bool(member_nat(B3),A2))
           => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A2)) ) )
     => ( ( A2 != bot_bot_set_nat )
       => pp(aa_set_nat_bool(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A2)),A2)) ) ) ).

% Gcd_in
tff(fact_9342_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_9343_abs__Gcd__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Gcd_int(K5)) = gcd_Gcd_int(K5) ).

% abs_Gcd_eq
tff(fact_9344_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U))) ).

% finite_greaterThanLessThan
tff(fact_9345_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9346_Gcd__dvd__int,axiom,
    ! [A: int,A2: set_int] :
      ( pp(aa_set_int_bool(member_int(A),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,gcd_Gcd_int(A2)),A)) ) ).

% Gcd_dvd_int
tff(fact_9347_Gcd__greatest__int,axiom,
    ! [A2: set_int,A: int] :
      ( ! [B3: int] :
          ( pp(aa_set_int_bool(member_int(B3),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B3)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),gcd_Gcd_int(A2))) ) ).

% Gcd_greatest_int
tff(fact_9348_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_9349_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5))) ).

% Gcd_int_greater_eq_0
tff(fact_9350_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_wb),Xa2),X)) ).

% times_int.abs_eq
tff(fact_9351_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X4: nat,Y3: nat] : Z2 != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y3)) ).

% eq_Abs_Integ
tff(fact_9352_int_Oabs__induct,axiom,
    ! [P: fun_int_bool,X: int] :
      ( ! [Y3: product_prod_nat_nat] : pp(aa_int_bool(P,aa_Pro4240683299535347697at_int(abs_Integ,Y3)))
     => pp(aa_int_bool(P,X)) ) ).

% int.abs_induct
tff(fact_9353_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_9354_int__def,axiom,
    ! [N2: nat] : aa_nat_int(semiri1314217659103216013at_int,N2) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N2),zero_zero_nat)) ).

% int_def
tff(fact_9355_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).

% nat.abs_eq
tff(fact_9356_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_wc),X)) ).

% uminus_int.abs_eq
tff(fact_9357_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_9358_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_we),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_9359_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_wg),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_9360_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_wi),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_9361_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_wk),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_9362_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_wg),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_9363_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_we),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_9364_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N2)) = aa_num_num(inc,aa_nat_num(num_of_nat,N2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N2)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_9365_num__of__nat__numeral__eq,axiom,
    ! [Q4: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q4)) = Q4 ).

% num_of_nat_numeral_eq
tff(fact_9366_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9367_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N2)) = N2 ) ) ).

% numeral_num_of_nat
tff(fact_9368_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),one_one_nat))
     => ( aa_nat_num(num_of_nat,N2) = one ) ) ).

% num_of_nat_One
tff(fact_9369_nat_Orep__eq,axiom,
    ! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).

% nat.rep_eq
tff(fact_9370_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N2)) ) ) ).

% num_of_nat_double
tff(fact_9371_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N2)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9372_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_wc)) ).

% uminus_int_def
tff(fact_9373_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(divide_divide_int(A),B))),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(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_9374_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% diff_rat
tff(fact_9375_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),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_9376_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(plus_plus_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% add_rat
tff(fact_9377_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),aa_int_rat(aa_int_fun_int_rat(fract,C),D)))
        <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),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_9378_eq__rat_I3_J,axiom,
    ! [A: int,C: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),A) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),C) ).

% eq_rat(3)
tff(fact_9379_eq__rat_I2_J,axiom,
    ! [A: int] : aa_int_rat(aa_int_fun_int_rat(fract,A),zero_zero_int) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% eq_rat(2)
tff(fact_9380_Rat__induct__pos,axiom,
    ! [P: fun_rat_bool,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3))) )
     => pp(aa_rat_bool(P,Q4)) ) ).

% Rat_induct_pos
tff(fact_9381_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(aa_int_fun_int_rat(fract,A),B) = aa_int_rat(aa_int_fun_int_rat(fract,C),D) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9382_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_rat(aa_int_fun_int_rat(fract,A),B) ) ) ).

% mult_rat_cancel
tff(fact_9383_rat__number__collapse_I6_J,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),zero_zero_int) = zero_zero_rat ).

% rat_number_collapse(6)
tff(fact_9384_rat__number__collapse_I1_J,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),K) = zero_zero_rat ).

% rat_number_collapse(1)
tff(fact_9385_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_9386_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).

% zero_less_Fract_iff
tff(fact_9387_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).

% Fract_less_zero_iff
tff(fact_9388_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).

% Fract_less_one_iff
tff(fact_9389_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).

% one_less_Fract_iff
tff(fact_9390_Fract__add__one,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M),N2)),N2) = aa_rat_rat(plus_plus_rat(aa_int_rat(aa_int_fun_int_rat(fract,M),N2)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9391_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).

% zero_le_Fract_iff
tff(fact_9392_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),zero_zero_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).

% Fract_le_zero_iff
tff(fact_9393_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).

% Fract_le_one_iff
tff(fact_9394_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).

% one_le_Fract_iff
tff(fact_9395_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_wb)) ).

% times_int_def
tff(fact_9396_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_wk)) ).

% minus_int_def
tff(fact_9397_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_wi)) ).

% plus_int_def
tff(fact_9398_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9399_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N2))) = case_option_int_num(zero_zero_int,aTP_Lamm_wm(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N2)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9400_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9401_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9402_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] : bit_take_bit_num(aa_nat_nat(suc,N2),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9403_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_9404_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_9405_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A2))) ).

% zero_notin_Suc_image
tff(fact_9406_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_we)) ).

% less_int_def
tff(fact_9407_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_wg)) ).

% less_eq_int_def
tff(fact_9408_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9409_image__Suc__lessThan,axiom,
    ! [N2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N2)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N2) ).

% image_Suc_lessThan
tff(fact_9410_image__Suc__atMost,axiom,
    ! [N2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N2)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N2)) ).

% image_Suc_atMost
tff(fact_9411_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N2)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N2))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9412_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N2)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9413_lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N2)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N2))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9414_atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N2)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N2))) ).

% atMost_Suc_eq_insert_0
tff(fact_9415_take__bit__num__def,axiom,
    ! [N2: nat,M: num] :
      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
       => ( bit_take_bit_num(N2,M) = none_num ) )
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
       => ( bit_take_bit_num(N2,M) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N2),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).

% take_bit_num_def
tff(fact_9416_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N2))) ).

% and_minus_numerals(7)
tff(fact_9417_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N2)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bitM,N2))) ).

% and_minus_numerals(3)
tff(fact_9418_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N2))) ).

% and_minus_numerals(8)
tff(fact_9419_Gcd__abs__eq,axiom,
    ! [K5: set_int] : gcd_Gcd_int(image_int_int(abs_abs_int,K5)) = gcd_Gcd_int(K5) ).

% Gcd_abs_eq
tff(fact_9420_Gcd__int__eq,axiom,
    ! [N5: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N5)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N5)) ).

% Gcd_int_eq
tff(fact_9421_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_wn,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9422_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N2),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N2,M))) ).

% take_bit_num_simps(4)
tff(fact_9423_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N2),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_wo,bit_take_bit_num(N2,M)) ).

% take_bit_num_simps(3)
tff(fact_9424_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N2)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N2))) ).

% and_minus_numerals(4)
tff(fact_9425_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastAtMost
tff(fact_9426_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastLessThan
tff(fact_9427_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_wp(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_9428_and__not__num__eq__None__iff,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num(M,N2) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N2))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9429_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9430_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N2)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N2,M)) ).

% int_numeral_not_and_num
tff(fact_9431_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N2))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N2)) ).

% int_numeral_and_not_num
tff(fact_9432_positive__rat,axiom,
    ! [A: int,B: int] :
      ( pp(aa_rat_bool(positive,aa_int_rat(aa_int_fun_int_rat(fract,A),B)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).

% positive_rat
tff(fact_9433_Rat_Opositive__zero,axiom,
    ~ pp(aa_rat_bool(positive,zero_zero_rat)) ).

% Rat.positive_zero
tff(fact_9434_Rat_Opositive__minus,axiom,
    ! [X: rat] :
      ( ~ pp(aa_rat_bool(positive,X))
     => ( ( X != zero_zero_rat )
       => pp(aa_rat_bool(positive,aa_rat_rat(uminus_uminus_rat,X))) ) ) ).

% Rat.positive_minus
tff(fact_9435_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
    <=> pp(aa_rat_bool(positive,aa_rat_rat(minus_minus_rat(Y),X))) ) ).

% less_rat_def
tff(fact_9436_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa_rat_bool(positive,X))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_9437_Gcd__int__def,axiom,
    ! [K5: set_int] : gcd_Gcd_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Gcd_int_def
tff(fact_9438_Rat_Opositive__def,axiom,
    positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_wq) ).

% Rat.positive_def
tff(fact_9439_measure__function__int,axiom,
    fun_is_measure_int(comp_int_nat_int(nat2,abs_abs_int)) ).

% measure_function_int
tff(fact_9440_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_9441_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),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))),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9442_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),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))),N2) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N2)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9443_min__strict__def,axiom,
    fun_min_strict = min_ex6901939911449802026at_nat(fun_pair_less) ).

% min_strict_def
tff(fact_9444_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gy,N2))) = N2 ) ) ).

% Max_divisors_self_nat
tff(fact_9445_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa_set_nat_bool(finite_finite_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U))) ).

% finite_greaterThanAtMost
tff(fact_9446_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_gx,N2))) = aa_int_int(abs_abs_int,N2) ) ) ).

% Max_divisors_self_int
tff(fact_9447_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_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_9448_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_9449_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N2) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_wr(N2),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9450_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S3)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S3)))) ) ).

% card_le_Suc_Max
tff(fact_9451_divide__nat__def,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = zero_zero_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N2) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ws(N2),M))) ) ) ) ).

% divide_nat_def
tff(fact_9452_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N2) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_wt(N2),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9453_min__weak__def,axiom,
    fun_min_weak = sup_su5525570899277871387at_nat(min_ex6901939911449802026at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_9454_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa_set_int_bool(finite_finite_int,set_or6656581121297822940st_int(L,U))) ).

% finite_greaterThanAtMost_int
tff(fact_9455_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_9456_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_9457_max__weak__def,axiom,
    fun_max_weak = sup_su5525570899277871387at_nat(max_ex8135407076693332796at_nat(fun_pair_leq),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_9458_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9459_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9460_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9461_max__strict__def,axiom,
    fun_max_strict = max_ex8135407076693332796at_nat(fun_pair_less) ).

% max_strict_def
tff(fact_9462_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
         => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_wu,M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9463_min__enat__simps_I3_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat),Q4) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9464_min__enat__simps_I2_J,axiom,
    ! [Q4: extended_enat] : aa_Ext7969882492699152700d_enat(ord_mi8085742599997312461d_enat(Q4),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9465_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)) ).

% min_Suc_Suc
tff(fact_9466_min__0L,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N2) = zero_zero_nat ).

% min_0L
tff(fact_9467_min__0R,axiom,
    ! [N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N2),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9468_min__numeral__Suc,axiom,
    ! [K: num,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N2)) ).

% min_numeral_Suc
tff(fact_9469_min__Suc__numeral,axiom,
    ! [N2: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N2)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N2),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9470_min__diff,axiom,
    ! [M: nat,I: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)),I) ).

% min_diff
tff(fact_9471_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N2)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2),Q4)) ).

% nat_mult_min_left
tff(fact_9472_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N2),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_min_right
tff(fact_9473_min__Suc2,axiom,
    ! [M: nat,N2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N2)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_wv,N2)),M) ).

% min_Suc2
tff(fact_9474_min__Suc1,axiom,
    ! [N2: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N2)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ww,N2)),M) ).

% min_Suc1
tff(fact_9475_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9476_Inf__real__def,axiom,
    ! [X5: set_real] : comple4887499456419720421f_real(X5) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X5))) ).

% Inf_real_def
tff(fact_9477_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_9478_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_9479_suminf__eq__SUP__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I2)))
       => ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_wx(X5),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9480_Sup__nat__def,axiom,
    ! [X5: set_nat] :
      ( ( ( X5 = bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X5) = zero_zero_nat ) )
      & ( ( X5 != bot_bot_set_nat )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,X5) = aa_set_nat_nat(lattic8265883725875713057ax_nat,X5) ) ) ) ).

% Sup_nat_def
tff(fact_9481_range__mod,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wy,N2),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N2) ) ) ).

% range_mod
tff(fact_9482_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_9483_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9484_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = insert_real(zero_zero_real,bot_bot_set_real) ) )
      & ( ( A != zero_zero_real )
       => ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = top_top_set_real ) ) ) ).

% range_mult
tff(fact_9485_infinite__UNIV__int,axiom,
    ~ pp(aa_set_int_bool(finite_finite_int,top_top_set_int)) ).

% infinite_UNIV_int
tff(fact_9486_surj__prod__decode,axiom,
    image_5846123807819985514at_nat(nat_prod_decode,top_top_set_nat) = top_to4669805908274784177at_nat ).

% surj_prod_decode
tff(fact_9487_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_9488_int__in__range__abs,axiom,
    ! [N2: nat] : pp(aa_set_int_bool(member_int(aa_nat_int(semiri1314217659103216013at_int,N2)),image_int_int(abs_abs_int,top_top_set_int))) ).

% int_in_range_abs
tff(fact_9489_root__def,axiom,
    ! [N2: nat,X: real] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N2),X) = zero_zero_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_real_real(aa_nat_fun_real_real(root,N2),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_wz,N2)),X) ) ) ) ).

% root_def
tff(fact_9490_DERIV__real__root__generic,axiom,
    ! [N2: nat,X: real,D4: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ( X != zero_zero_real )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N2),D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9491_DERIV__even__real__root,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N2),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,N2))),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9492_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
       => ? [Z: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),B))
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F4,Z)) ) ) ) ) ).

% MVT2
tff(fact_9493_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y4),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9494_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y4)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9495_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ( ! [X4: real] :
            ( member_real(X4,set_or1222579329274155063t_real(A,B))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X4))) )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9496_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9497_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9498_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9499_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,X: real,Y: real] :
      ( ! [X4: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real))
     => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ).

% DERIV_isconst_all
tff(fact_9500_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9501_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9502_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( member_real(X,set_or1633881224788618240n_real(A,B))
       => ( member_real(Y,set_or1633881224788618240n_real(A,B))
         => ( ! [X4: real] :
                ( member_real(X4,set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) )
           => ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9503_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => ( aa_real_real(F,X) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9504_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9505_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                 => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9506_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9507_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9508_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9509_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,X: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S3))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [D5: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
            & ! [H3: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3),S3)
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D5))
                   => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9510_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,X))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9511_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
       => ( ! [Y3: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y3))) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9512_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9513_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,X: real,N2: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_xa(G),N2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(aa_real_real(G,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9514_DERIV__pow,axiom,
    ! [N2: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_xb,N2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_9515_has__real__derivative__powr,axiom,
    ! [Z2: real,R2: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z2))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_xc,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9516_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9517_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_xd(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9518_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_xe(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(F,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9519_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9520_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
         => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
           => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9521_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_xf(F),X4)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
         => has_fi5821293074295781190e_real(aTP_Lamm_xh(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_xf(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9522_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9523_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_9524_DERIV__real__root,axiom,
    ! [N2: nat,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N2),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9525_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9526_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(power_power_real(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9527_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N2: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M5: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T2: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xi(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9528_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N2: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M5: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xi(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9529_DERIV__odd__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N2),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_nat_real(power_power_real(aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9530_Maclaurin,axiom,
    ! [H: real,N2: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M5: nat,T2: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),H))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kd(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(H),N2))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9531_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N2: nat] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ? [T2: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H))
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kd(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(H),N2))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9532_Maclaurin__minus,axiom,
    ! [H: real,N2: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M5: nat,T2: real] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),zero_zero_real)) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),zero_zero_real))
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_kd(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(H),N2))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9533_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N2: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => ( ( X != zero_zero_real )
         => ( ! [M5: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T2: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2)))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xi(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9534_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N2: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M5: nat,T2: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
       => ? [T2: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xi(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(X),N2))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9535_Taylor,axiom,
    ! [N2: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
                 => ( ( X != C )
                   => ? [T2: real] :
                        ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T2))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),C)) ) )
                        & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
                         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T2))
                            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),X)) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xj(Diff),C),X)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),C)),N2))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9536_Taylor__up,axiom,
    ! [N2: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
             => ? [T2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),B))
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xk(Diff),B),C)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),N2))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9537_Taylor__down,axiom,
    ! [N2: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M5: nat,T2: real] :
              ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T2))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),B)) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
             => ? [T2: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T2))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T2),C))
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xk(Diff),A),C)),aa_nat_set_nat(set_ord_lessThan_nat,N2))),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,N2),T2)),aa_nat_real(semiri2265585572941072030t_real,N2))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),N2))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9538_Maclaurin__lemma2,axiom,
    ! [N2: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
      ( ! [M5: nat,T2: real] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M5),N2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T2))
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T2),H)) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M5),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M5)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
     => ( ( N2 = aa_nat_nat(suc,K) )
       => ! [M2: nat,T5: real] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T5))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T5),H)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_xm(N2),Diff),B2),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xn(Diff),M2),T5)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9539_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => has_fi5821293074295781190e_real(aTP_Lamm_xo,suminf_real(aTP_Lamm_xp(X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9540_surj__int__encode,axiom,
    image_int_nat(nat_int_encode,top_top_set_int) = top_top_set_nat ).

% surj_int_encode
tff(fact_9541_int__encode__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_nat(nat_int_encode,X) = aa_int_nat(nat_int_encode,Y) )
    <=> ( X = Y ) ) ).

% int_encode_eq
tff(fact_9542_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X2))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X2))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9543_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X2))),R3)) )
               => ( aa_real_real(F,X2) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9544_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
       => ? [R3: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X2))),R3)) )
               => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X2)),zero_zero_real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9545_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
       => ( ! [Z: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),B))
               => ( aa_real_real(G,aa_real_real(F,Z)) = Z ) ) )
         => ( ! [Z: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),B))
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9546_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_9547_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9548_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_xq,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9549_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D4: real,G: fun_real_real,X: real,A: real,B: real] :
      ( has_fi5821293074295781190e_real(F,D4,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
           => ( ! [Y3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y3))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B))
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9550_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9551_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9552_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
     => ( ! [X4: real] :
            ( member_real(X4,set_or1633881224788618240n_real(B,X))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4))) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ).

% LIM_less_bound
tff(fact_9553_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9554_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
     => ( ! [Z: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),X))),D))
           => ( aa_real_real(G,aa_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),X))),D))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9555_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [Z: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),B))
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) ) )
       => ( ! [Z: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),B))
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),G) ) )
         => ( ! [Z: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),B))
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z),topolo2177554685111907308n_real(Z,top_top_set_real)) ) )
           => ( ! [Z: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z))
                 => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),B))
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z),topolo2177554685111907308n_real(Z,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F4,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9556_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real))
         => ! [N6: nat] : member_real(suminf_real(aTP_Lamm_xr(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N6)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N6))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9557_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat)))
         => ! [N6: nat] : member_real(suminf_real(aTP_Lamm_xr(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N6))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N6)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9558_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N)))
       => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N)))
         => filterlim_nat_real(aTP_Lamm_xs(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xr(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9559_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xt,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9560_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9561_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N))))
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N)))
       => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N)))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xu(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N6)),L3))
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_nat_real(G,N6)))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9562_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X5,N))) )
     => filterlim_nat_real(aTP_Lamm_xv(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9563_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_xw,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9564_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
     => filterlim_nat_real(aTP_Lamm_xx(C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9565_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_xy,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9566_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_xz(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9567_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N))))
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),L))
       => ( ! [E: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
             => ? [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N6)),E))) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9568_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
       => filterlim_nat_real(power_power_real(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9569_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ya(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9570_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(power_power_real(aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_9571_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real))
     => filterlim_nat_real(power_power_real(C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9572_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
     => filterlim_nat_real(aTP_Lamm_yb(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9573_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_yc(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9574_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aTP_Lamm_yd(R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9575_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_xr(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9576_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N)))
       => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N)))
         => summable_real(aTP_Lamm_xr(A)) ) ) ) ).

% summable
tff(fact_9577_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_ye(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_yf(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9578_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_yg(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xr(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9579_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
     => filterlim_nat_real(aTP_Lamm_by(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9580_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N)))
       => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N)))
         => filterlim_nat_real(aTP_Lamm_yg(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xr(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9581_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,N2: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N)))
       => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N2)))),suminf_real(aTP_Lamm_xr(A)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9582_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N)))
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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] :
              ( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N6)))),L3))
              & filterlim_nat_real(aTP_Lamm_yg(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N6)),one_one_nat)))))
              & filterlim_nat_real(aTP_Lamm_xs(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9583_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_xs(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_xr(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9584_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,N2: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N)))
       => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_xr(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(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))),N2)),one_one_nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9585_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9586_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yh,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9587_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_9588_filterlim__pow__at__bot__odd,axiom,
    ! [N2: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yi(N2),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9589_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9590_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_yj(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9591_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_yk(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9592_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_9593_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N4))
         => pp(aa_nat_bool(P,N4)) ) ) ).

% eventually_sequentially
tff(fact_9594_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X4))
         => pp(aa_nat_bool(P,X4)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9595_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F3) ) ).

% le_sequentially
tff(fact_9596_sequentially__offset,axiom,
    ! [P: fun_nat_bool,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_yk(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9597_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_yl(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9598_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_9599_eventually__at__right__to__0,axiom,
    ! [P: fun_real_bool,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ym(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_9600_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_yl(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9601_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_9602_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9603_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_9604_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_9605_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_9606_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_9607_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_9608_lhopital,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aTP_Lamm_yn(G),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),F3,topolo2177554685111907308n_real(X,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yq(F),G),F3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_9609_lhopital__left,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aTP_Lamm_yn(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yq(F),G),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_9610_lhopital__right,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aTP_Lamm_yn(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yq(F),G),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_9611_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F4: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_yn(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F0),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yq(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_9612_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9613_eventually__at__right__to__top,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_yr(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_9614_eventually__at__top__to__right,axiom,
    ! [P: fun_real_bool] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_yr(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_9615_Bseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => bfun_nat_real(power_power_real(X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9616_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_ys,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9617_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_9618_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_9619_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_yt,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9620_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9621_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_yn(G2),at_top_real)
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),at_top_real)
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),topolo2815343760600316023s_real(X),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9622_lhopital__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9623_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X4))
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9624_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_9625_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_yn(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(F),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G2),F4),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_9626_filterlim__pow__at__bot__even,axiom,
    ! [N2: nat,F: fun_real_real,F3: filter_real] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yi(N2),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9627_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_bool,B: nat] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_ex_nat
tff(fact_9628_Greatest__le__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).

% Greatest_le_nat
tff(fact_9629_GreatestI__nat,axiom,
    ! [P: fun_nat_bool,K: nat,B: nat] :
      ( pp(aa_nat_bool(P,K))
     => ( ! [Y3: nat] :
            ( pp(aa_nat_bool(P,Y3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),B)) )
       => pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).

% GreatestI_nat
tff(fact_9630_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X4: vEBT_VEBT] :
                    ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                   => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9631_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                     => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                  & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                  & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9632_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList2,Summary)),Deg4))
    <=> ( ( Deg = Deg4 )
        & ! [X3: vEBT_VEBT] :
            ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
           => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
        & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
        & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Deg),TreeList2),Summary)),Mima2) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9633_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9634_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
                       => pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                    & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                    & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9635_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Deg2),TreeList),Summary2)),Mima) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9636_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
      <=> pp(Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
                         => pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
                      & case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Deg2),TreeList),Summary2)),Mima) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList,Summary2),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9637_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ( ! [X4: real] :
              ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
         => ( ! [X4: real] :
                ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
           => ( ! [X4: real] :
                  ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
                    & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B)) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X4,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C3))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C3),B))
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9638_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ? [L3: real,Z: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z))
              & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),B))
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z,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_9639_Rolle__deriv,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),B))
                & ! [X2: real] : aa_real_real(aa_rea7340605660300375871l_real(F4,Z),X2) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9640_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Xi))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi),B))
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F4,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9641_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y4)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9642_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ! [X4: real] :
            ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y4),zero_zero_real)) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9643_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9644_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
             => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
           => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9645_Rolle,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
     => ( ( aa_real_real(F,A) = aa_real_real(F,B) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
         => ( ! [X4: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X4))
               => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B))
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
           => ? [Z: real] :
                ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z))
                & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),B))
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9646_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_9647_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9648_mono__times__nat,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N2)) ) ).

% mono_times_nat
tff(fact_9649_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yv,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9650_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_wz,N2),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9651_log__inj,axiom,
    ! [B: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9652_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_yw) ).

% Rats_eq_int_div_nat
tff(fact_9653_surj__int__decode,axiom,
    image_nat_int(nat_int_decode,top_top_set_nat) = top_top_set_int ).

% surj_int_decode
tff(fact_9654_Rats__abs__iff,axiom,
    ! [X: real] :
      ( member_real(aa_real_real(abs_abs_real,X),field_5140801741446780682s_real)
    <=> member_real(X,field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_9655_int__decode__inverse,axiom,
    ! [N2: nat] : aa_int_nat(nat_int_encode,aa_nat_int(nat_int_decode,N2)) = N2 ).

% int_decode_inverse
tff(fact_9656_int__encode__inverse,axiom,
    ! [X: int] : aa_nat_int(nat_int_decode,aa_int_nat(nat_int_encode,X)) = X ).

% int_encode_inverse
tff(fact_9657_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N: nat] :
          ( pp(aa_set_nat_bool(member_nat(N),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) )
     => pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,K)),N5)) ) ).

% inj_on_diff_nat
tff(fact_9658_inj__prod__encode,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(inj_on2178005380612969504at_nat(nat_prod_encode),A2)) ).

% inj_prod_encode
tff(fact_9659_inj__prod__decode,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(inj_on5538052773655684606at_nat(nat_prod_decode),A2)) ).

% inj_prod_decode
tff(fact_9660_inj__Suc,axiom,
    ! [N5: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N5)) ).

% inj_Suc
tff(fact_9661_inj__int__decode,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(inj_on_nat_int(nat_int_decode),A2)) ).

% inj_int_decode
tff(fact_9662_inj__int__encode,axiom,
    ! [A2: set_int] : pp(aa_set_int_bool(inj_on_int_nat(nat_int_encode),A2)) ).

% inj_int_encode
tff(fact_9663_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( member_real(X4,field_5140801741446780682s_real)
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),X4)) ) ).

% Rats_no_top_le
tff(fact_9664_int__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_int(nat_int_decode,X) = aa_nat_int(nat_int_decode,Y) )
    <=> ( X = Y ) ) ).

% int_decode_eq
tff(fact_9665_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( member_real(X4,field_5140801741446780682s_real)
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X)) ) ).

% Rats_no_bot_less
tff(fact_9666_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [X4: real] :
          ( member_real(X4,field_5140801741446780682s_real)
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X4))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_9667_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_9668_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_9669_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F))) ) ) ) ).

% suminf_reindex_mono
tff(fact_9670_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_yx) ).

% Rats_eq_int_div_int
tff(fact_9671_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
       => ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
         => ( ! [X4: nat] :
                ( ~ pp(aa_set_nat_bool(member_nat(X4),image_nat_nat(G,top_top_set_nat)))
               => ( aa_nat_real(F,X4) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9672_powr__real__of__int_H,axiom,
    ! [X: real,N2: int] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( ( ( X != zero_zero_real )
          | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N2)) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N2)) = aa_int_real(power_int_real(X),N2) ) ) ) ).

% powr_real_of_int'
tff(fact_9673_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_yy)) ).

% pred_nat_def
tff(fact_9674_pow_Osimps_I1_J,axiom,
    ! [X: num] : aa_num_num(pow(X),one) = X ).

% pow.simps(1)
tff(fact_9675_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : aa_num_num(pow(X),aa_num_num(bit1,Y)) = aa_num_num(times_times_num(aa_num_num(sqr,aa_num_num(pow(X),Y))),X) ).

% pow.simps(3)
tff(fact_9676_sqr_Osimps_I2_J,axiom,
    ! [N2: num] : aa_num_num(sqr,aa_num_num(bit0,N2)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,N2))) ).

% sqr.simps(2)
tff(fact_9677_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9678_sqr__conv__mult,axiom,
    ! [X: num] : aa_num_num(sqr,X) = aa_num_num(times_times_num(X),X) ).

% sqr_conv_mult
tff(fact_9679_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : aa_num_num(pow(X),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(pow(X),Y)) ).

% pow.simps(2)
tff(fact_9680_sqr_Osimps_I3_J,axiom,
    ! [N2: num] : aa_num_num(sqr,aa_num_num(bit1,N2)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(sqr,N2)),N2))) ).

% sqr.simps(3)
tff(fact_9681_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F4: fun_real_real] :
      ( ! [X4: real] : has_fi5821293074295781190e_real(F,aa_real_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real))
     => ( ! [X4: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F4,X4)))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9682_Arg__bounded,axiom,
    ! [Z2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z2)))
      & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z2)),pi)) ) ).

% Arg_bounded
tff(fact_9683_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_yz(N2),aa_nat_set_nat(set_ord_lessThan_nat,N2)),collect_complex(aTP_Lamm_bq(N2)))) ) ).

% bij_betw_roots_unity
tff(fact_9684_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9685_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,N2: nat] :
      ( order_5726023648592871131at_nat(F)
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2))) ) ).

% strict_mono_imp_increasing
tff(fact_9686_cis__Arg,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_real_complex(cis,aa_complex_real(arg,Z2)) = aa_complex_complex(sgn_sgn_complex,Z2) ) ) ).

% cis_Arg
tff(fact_9687_cis__neq__zero,axiom,
    ! [A: real] : aa_real_complex(cis,A) != zero_zero_complex ).

% cis_neq_zero
tff(fact_9688_cis__Arg__unique,axiom,
    ! [Z2: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z2) = aa_real_complex(cis,X) )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
         => ( aa_complex_real(arg,Z2) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_9689_Arg__correct,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z2) = aa_real_complex(cis,aa_complex_real(arg,Z2)) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z2)))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z2)),pi)) ) ) ).

% Arg_correct
tff(fact_9690_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_9691_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4)))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_za(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9692_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
       => pp(aa_set_complex_bool(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N2),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,N2))))),collect_complex(aTP_Lamm_bq(N2))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_vy(C),N2)))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9693_Arg__def,axiom,
    ! [Z2: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( aa_complex_real(arg,Z2) = zero_zero_real ) )
      & ( ( Z2 != zero_zero_complex )
       => ( aa_complex_real(arg,Z2) = fChoice_real(aTP_Lamm_zb(Z2)) ) ) ) ).

% Arg_def
tff(fact_9694_bij__betw__Suc,axiom,
    ! [M7: set_nat,N5: set_nat] :
      ( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M7),N5))
    <=> ( image_nat_nat(suc,M7) = N5 ) ) ).

% bij_betw_Suc
tff(fact_9695_bij__int__decode,axiom,
    pp(aa_set_int_bool(bij_betw_nat_int(nat_int_decode,top_top_set_nat),top_top_set_int)) ).

% bij_int_decode
tff(fact_9696_bij__int__encode,axiom,
    pp(aa_set_nat_bool(bij_betw_int_nat(nat_int_encode,top_top_set_int),top_top_set_nat)) ).

% bij_int_encode
tff(fact_9697_bij__prod__decode,axiom,
    pp(aa_set3135850010578026034t_bool(bij_be8693218025023041337at_nat(nat_prod_decode,top_top_set_nat),top_to4669805908274784177at_nat)) ).

% bij_prod_decode
tff(fact_9698_bij__prod__encode,axiom,
    pp(aa_set_nat_bool(bij_be5333170631980326235at_nat(nat_prod_encode,top_to4669805908274784177at_nat),top_top_set_nat)) ).

% bij_prod_encode
tff(fact_9699_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_zc(X)) ).

% quotient_of_def
tff(fact_9700_normalize__stable,axiom,
    ! [Q4: int,P3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q4))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P3),Q4))
       => ( normalize(product_Pair_int_int(P3,Q4)) = product_Pair_int_int(P3,Q4) ) ) ) ).

% normalize_stable
tff(fact_9701_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),D))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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_9702_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3)) ) ) ).

% Rat_cases
tff(fact_9703_Rat__induct,axiom,
    ! [P: fun_rat_bool,Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3))
           => pp(aa_rat_bool(P,aa_int_rat(aa_int_fun_int_rat(fract,A3),B3))) ) )
     => pp(aa_rat_bool(P,Q4)) ) ).

% Rat_induct
tff(fact_9704_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q4 = aa_int_rat(aa_int_fun_int_rat(fract,A3),B3) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
           => ( ( A3 != zero_zero_int )
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A3),B3)) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9705_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X4: product_prod_int_int] :
      ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(X4)),product_snd_int_int(X4)) )
      & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(X4)))
      & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(X4)),product_snd_int_int(X4)))
      & ! [Y4: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Y4)))
            & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Y4)),product_snd_int_int(Y4))) )
         => ( Y4 = X4 ) ) ) ).

% quotient_of_unique
tff(fact_9706_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),D))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9707_coprime__Suc__0__left,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N2)) ).

% coprime_Suc_0_left
tff(fact_9708_coprime__Suc__0__right,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N2),aa_nat_nat(suc,zero_zero_nat))) ).

% coprime_Suc_0_right
tff(fact_9709_coprime__Suc__right__nat,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N2),aa_nat_nat(suc,N2))) ).

% coprime_Suc_right_nat
tff(fact_9710_coprime__Suc__left__nat,axiom,
    ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,N2)),N2)) ).

% coprime_Suc_left_nat
tff(fact_9711_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),A))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),B))
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_9712_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)),N2)) ) ).

% coprime_diff_one_left_nat
tff(fact_9713_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_9714_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( member_real(X,field_5140801741446780682s_real)
     => ~ ! [M5: nat,N: nat] :
            ( ( N != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M5)),aa_nat_real(semiri5074537144036343181t_real,N)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M5),N)) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9715_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_9716_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_gcd_int,Xs),zero_zero_int) ).

% Gcd_int_set_eq_fold
tff(fact_9717_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
     => ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_zd(M),N5))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ze(M),N5)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_zf(M),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9718_card__length__sum__list,axiom,
    ! [M: nat,N5: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_zd(M),N5))) = 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,N5),M)),one_one_nat)),N5) ).

% card_length_sum_list
tff(fact_9719_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_zg(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_zh(M)) ).

% pairs_le_eq_Sigma
tff(fact_9720_take__upt,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),N2))
     => ( take_nat(M,upt(I,N2)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_9721_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J))
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9722_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
     => ( groups4561878855575611511st_nat(upt(M,N2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_au),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N2)) ) ) ).

% sum_list_upt
tff(fact_9723_map__Suc__upt,axiom,
    ! [M: nat,N2: nat] : map_nat_nat(suc,upt(M,N2)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N2)) ).

% map_Suc_upt
tff(fact_9724_map__add__upt,axiom,
    ! [N2: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zi,N2),upt(zero_zero_nat,M)) = upt(N2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% map_add_upt
tff(fact_9725_map__decr__upt,axiom,
    ! [M: nat,N2: nat] : map_nat_nat(aTP_Lamm_zj,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N2))) = upt(M,N2) ).

% map_decr_upt
tff(fact_9726_atLeastAtMost__upt,axiom,
    ! [N2: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N2),M) = set_nat2(upt(N2,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9727_atLeast__upt,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N2) = set_nat2(upt(zero_zero_nat,N2)) ).

% atLeast_upt
tff(fact_9728_greaterThanAtMost__upt,axiom,
    ! [N2: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N2),M) = set_nat2(upt(aa_nat_nat(suc,N2),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_9729_greaterThanLessThan__upt,axiom,
    ! [N2: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N2),M) = set_nat2(upt(aa_nat_nat(suc,N2),M)) ).

% greaterThanLessThan_upt
tff(fact_9730_atMost__upto,axiom,
    ! [N2: nat] : aa_nat_set_nat(set_ord_atMost_nat,N2) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N2))) ).

% atMost_upto
tff(fact_9731_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N2))) ).

% sorted_wrt_upt
tff(fact_9732_sorted__upt,axiom,
    ! [M: nat,N2: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N2))) ).

% sorted_upt
tff(fact_9733_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_9734_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list_nat,Q4: nat] :
      ( ( cons_nat(M,cons_nat(N2,Ns)) = upt(M,Q4) )
    <=> ( cons_nat(N2,Ns) = upt(aa_nat_nat(suc,M),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9735_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9736_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9737_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9738_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9739_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9740_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_9741_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9742_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9743_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N2)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N2)))
       => ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N2)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_9744_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X4: nat,Xs2: list_nat] : X != cons_nat(X4,Xs2) ) ).

% list_encode.cases
tff(fact_9745_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_9746_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9747_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9748_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9749_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_9750_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).

% drop_upt
tff(fact_9751_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = Y )
     => ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),X))
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat)) ) )
         => ~ ! [N: nat] :
                ( ( X = aa_nat_nat(suc,N) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_zk,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) )
                 => ~ pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_9752_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat(nat_list_decode,top_top_set_nat,top_top_set_list_nat) ).

% bij_list_decode
tff(fact_9753_list__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = aa_nat_list_nat(nat_list_decode,Y) )
    <=> ( X = Y ) ) ).

% list_decode_eq
tff(fact_9754_inj__list__decode,axiom,
    ! [A2: set_nat] : pp(aa_set_nat_bool(inj_on_nat_list_nat(nat_list_decode),A2)) ).

% inj_list_decode
tff(fact_9755_surj__list__decode,axiom,
    image_nat_list_nat(nat_list_decode,top_top_set_nat) = top_top_set_list_nat ).

% surj_list_decode
tff(fact_9756_list__decode_Osimps_I1_J,axiom,
    aa_nat_list_nat(nat_list_decode,zero_zero_nat) = nil_nat ).

% list_decode.simps(1)
tff(fact_9757_list__decode_Osimps_I2_J,axiom,
    ! [N2: nat] : aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N2)) = produc2761476792215241774st_nat(aTP_Lamm_zk,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ).

% list_decode.simps(2)
tff(fact_9758_list__decode_Opsimps_I1_J,axiom,
    ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),zero_zero_nat))
   => ( aa_nat_list_nat(nat_list_decode,zero_zero_nat) = nil_nat ) ) ).

% list_decode.psimps(1)
tff(fact_9759_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N: nat] :
              ( ( X = aa_nat_nat(suc,N) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_zk,aa_nat7603221163087113075at_nat(nat_prod_decode,N)) ) ) ) ) ).

% list_decode.elims
tff(fact_9760_list__decode_Opsimps_I2_J,axiom,
    ! [N2: nat] :
      ( pp(aa_nat_bool(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)))
     => ( aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N2)) = produc2761476792215241774st_nat(aTP_Lamm_zk,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ).

% list_decode.psimps(2)
tff(fact_9761_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X4,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9762_list__encode__inverse,axiom,
    ! [X: list_nat] : aa_nat_list_nat(nat_list_decode,aa_list_nat_nat(nat_list_encode,X)) = X ).

% list_encode_inverse
tff(fact_9763_list__decode__inverse,axiom,
    ! [N2: nat] : aa_list_nat_nat(nat_list_encode,aa_nat_list_nat(nat_list_decode,N2)) = N2 ).

% list_decode_inverse
tff(fact_9764_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_9765_inj__list__encode,axiom,
    ! [A2: set_list_nat] : inj_on_list_nat_nat(nat_list_encode,A2) ).

% inj_list_encode
tff(fact_9766_list__encode__eq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = aa_list_nat_nat(nat_list_encode,Y) )
    <=> ( X = Y ) ) ).

% list_encode_eq
tff(fact_9767_bij__list__encode,axiom,
    pp(aa_set_nat_bool(bij_be8532844293280997160at_nat(nat_list_encode,top_top_set_list_nat),top_top_set_nat)) ).

% bij_list_encode
tff(fact_9768_list__encode_Osimps_I1_J,axiom,
    aa_list_nat_nat(nat_list_encode,nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9769_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : aa_list_nat_nat(nat_list_encode,cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_list_nat_nat(nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_9770_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_9771_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9772_tl__upt,axiom,
    ! [M: nat,N2: nat] : tl_nat(upt(M,N2)) = upt(aa_nat_nat(suc,M),N2) ).

% tl_upt
tff(fact_9773_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( aa_list_nat_nat(nat_list_encode,X) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X4,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),aa_list_nat_nat(nat_list_encode,Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X4,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9774_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
tff(fact_9775_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9776_cauchy__def,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [M6: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M6))
             => ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M6)),aa_nat_rat(X5,N4)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_9777_cauchy__minus,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_zl,X5)) ) ).

% cauchy_minus
tff(fact_9778_cauchy__const,axiom,
    ! [X: rat] : cauchy(aTP_Lamm_zm(X)) ).

% cauchy_const
tff(fact_9779_cauchy__add,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zn,X5),Y5)) ) ) ).

% cauchy_add
tff(fact_9780_cauchy__diff,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_zo(X5),Y5)) ) ) ).

% cauchy_diff
tff(fact_9781_cauchy__mult,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zp,X5),Y5)) ) ) ).

% cauchy_mult
tff(fact_9782_cauchy__imp__bounded,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ? [B3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
          & ! [N6: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N6))),B3)) ) ) ).

% cauchy_imp_bounded
tff(fact_9783_cauchyD,axiom,
    ! [X5: fun_nat_rat,R2: rat] :
      ( cauchy(X5)
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [M2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M2))
           => ! [N6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M2)),aa_nat_rat(X5,N6)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_9784_cauchyI,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K4: nat] :
            ! [M5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),M5))
             => ! [N: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M5)),aa_nat_rat(X5,N)))),R3)) ) ) )
     => cauchy(X5) ) ).

% cauchyI
tff(fact_9785_le__Real,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)))
        <=> ! [R5: rat] :
              ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
                 => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N4)),aa_rat_rat(plus_plus_rat(aa_nat_rat(Y5,N4)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_9786_cauchy__not__vanishes,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ pp(vanishes(X5))
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
              ! [N6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N6)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9787_vanishes__const,axiom,
    ! [C: rat] :
      ( pp(vanishes(aTP_Lamm_zm(C)))
    <=> ( C = zero_zero_rat ) ) ).

% vanishes_const
tff(fact_9788_eq__Real,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => ( ( aa_fun_nat_rat_real(real2,X5) = aa_fun_nat_rat_real(real2,Y5) )
        <=> pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_zo(X5),Y5))) ) ) ) ).

% eq_Real
tff(fact_9789_inverse__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ( pp(vanishes(X5))
         => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X5)) = zero_zero_real ) )
        & ( ~ pp(vanishes(X5))
         => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X5)) = aa_fun_nat_rat_real(real2,aTP_Lamm_zq(X5)) ) ) ) ) ).

% inverse_Real
tff(fact_9790_Real__induct,axiom,
    ! [P: fun_real_bool,X: real] :
      ( ! [X7: fun_nat_rat] :
          ( cauchy(X7)
         => pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,X7))) )
     => pp(aa_real_bool(P,X)) ) ).

% Real_induct
tff(fact_9791_mult__Real,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zp,X5),Y5)) ) ) ) ).

% mult_Real
tff(fact_9792_vanishes__add,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( pp(vanishes(X5))
     => ( pp(vanishes(Y5))
       => pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zn,X5),Y5))) ) ) ).

% vanishes_add
tff(fact_9793_vanishes__minus,axiom,
    ! [X5: fun_nat_rat] :
      ( pp(vanishes(X5))
     => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_zl,X5))) ) ).

% vanishes_minus
tff(fact_9794_of__nat__Real,axiom,
    ! [X: nat] : aa_nat_real(semiri5074537144036343181t_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_zr(X)) ).

% of_nat_Real
tff(fact_9795_vanishes__diff,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( pp(vanishes(X5))
     => ( pp(vanishes(Y5))
       => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_zo(X5),Y5))) ) ) ).

% vanishes_diff
tff(fact_9796_zero__real__def,axiom,
    zero_zero_real = aa_fun_nat_rat_real(real2,aTP_Lamm_zs) ).

% zero_real_def
tff(fact_9797_one__real__def,axiom,
    one_one_real = aa_fun_nat_rat_real(real2,aTP_Lamm_zt) ).

% one_real_def
tff(fact_9798_of__int__Real,axiom,
    ! [X: int] : aa_int_real(ring_1_of_int_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_zu(X)) ).

% of_int_Real
tff(fact_9799_cauchy__inverse,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ pp(vanishes(X5))
       => cauchy(aTP_Lamm_zq(X5)) ) ) ).

% cauchy_inverse
tff(fact_9800_minus__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_zl,X5)) ) ) ).

% minus_Real
tff(fact_9801_add__Real,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zn,X5),Y5)) ) ) ) ).

% add_Real
tff(fact_9802_diff__Real,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_zo(X5),Y5)) ) ) ) ).

% diff_Real
tff(fact_9803_vanishes__mult__bounded,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( ? [A7: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A7))
          & ! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N))),A7)) )
     => ( pp(vanishes(Y5))
       => pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zp,X5),Y5))) ) ) ).

% vanishes_mult_bounded
tff(fact_9804_vanishes__diff__inverse,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ pp(vanishes(X5))
       => ( cauchy(Y5)
         => ( ~ pp(vanishes(Y5))
           => ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_zo(X5),Y5)))
             => pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_zv(X5),Y5))) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_9805_vanishes__def,axiom,
    ! [X5: fun_nat_rat] :
      ( pp(vanishes(X5))
    <=> ! [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
         => ? [K3: nat] :
            ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N4))),R5)) ) ) ) ).

% vanishes_def
tff(fact_9806_vanishesI,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [R3: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
         => ? [K4: nat] :
            ! [N: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N))),R3)) ) )
     => pp(vanishes(X5)) ) ).

% vanishesI
tff(fact_9807_vanishesD,axiom,
    ! [X5: fun_nat_rat,R2: rat] :
      ( pp(vanishes(X5))
     => ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R2))
       => ? [K2: nat] :
          ! [N6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
           => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N6))),R2)) ) ) ) ).

% vanishesD
tff(fact_9808_cauchy__not__vanishes__cases,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ pp(vanishes(X5))
       => ? [B3: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B3))
            & ? [K2: nat] :
                ( ! [N6: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X5,N6)))) )
                | ! [N6: nat] :
                    ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N6))
                   => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B3),aa_nat_rat(X5,N6))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9809_not__positive__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X5)))
      <=> ! [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
           => ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N4)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_9810_positive__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X5)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X5,N4))) ) ) ) ) ).

% positive_Real
tff(fact_9811_Real_Opositive__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(positive2,X))
     => ( pp(aa_real_bool(positive2,Y))
       => pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))) ) ) ).

% Real.positive_mult
tff(fact_9812_Real_Opositive__zero,axiom,
    ~ pp(aa_real_bool(positive2,zero_zero_real)) ).

% Real.positive_zero
tff(fact_9813_Real_Opositive__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(positive2,X))
     => ( pp(aa_real_bool(positive2,Y))
       => pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).

% Real.positive_add
tff(fact_9814_Real_Opositive__minus,axiom,
    ! [X: real] :
      ( ~ pp(aa_real_bool(positive2,X))
     => ( ( X != zero_zero_real )
       => pp(aa_real_bool(positive2,aa_real_real(uminus_uminus_real,X))) ) ) ).

% Real.positive_minus
tff(fact_9815_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
    <=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X))) ) ).

% less_real_def
tff(fact_9816_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X),N4))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9817_inverse__real_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,if_fun_nat_rat(vanishes(X),aTP_Lamm_zs,aTP_Lamm_zq(X))) ) ) ).

% inverse_real.abs_eq
tff(fact_9818_realrel__refl,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X5),X5)) ) ).

% realrel_refl
tff(fact_9819_one__real_Orsp,axiom,
    pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_zt),aTP_Lamm_zt)) ).

% one_real.rsp
tff(fact_9820_zero__real_Orsp,axiom,
    pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_zs),aTP_Lamm_zs)) ).

% zero_real.rsp
tff(fact_9821_real_Oabs__induct,axiom,
    ! [P: fun_real_bool,X: real] :
      ( ! [Y3: fun_nat_rat] :
          ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Y3),Y3))
         => pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,Y3))) )
     => pp(aa_real_bool(P,X)) ) ).

% real.abs_induct
tff(fact_9822_uminus__real_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_zl,X)) ) ) ).

% uminus_real.abs_eq
tff(fact_9823_plus__real_Oabs__eq,axiom,
    ! [Xa2: fun_nat_rat,X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa2),Xa2))
     => ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,Xa2)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zn,Xa2),X)) ) ) ) ).

% plus_real.abs_eq
tff(fact_9824_times__real_Oabs__eq,axiom,
    ! [Xa2: fun_nat_rat,X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa2),Xa2))
     => ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,Xa2)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zp,Xa2),X)) ) ) ) ).

% times_real.abs_eq
tff(fact_9825_realrelI,axiom,
    ! [X5: fun_nat_rat,Y5: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_zo(X5),Y5)))
         => pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X5),Y5)) ) ) ) ).

% realrelI
tff(fact_9826_realrel__def,axiom,
    ! [X2: fun_nat_rat,Xa: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X2),Xa))
    <=> ( cauchy(X2)
        & cauchy(Xa)
        & pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_zo(X2),Xa))) ) ) ).

% realrel_def
tff(fact_9827_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
     => ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X)))
      <=> ? [R5: rat] :
            ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
            & ? [K3: nat] :
              ! [N4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
               => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X,N4))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_9828_Real_Opositive__def,axiom,
    positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_zw) ).

% Real.positive_def
tff(fact_9829_inverse__real__def,axiom,
    inverse_inverse_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_zx) ).

% inverse_real_def
tff(fact_9830_uminus__real__def,axiom,
    uminus_uminus_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_zl) ).

% uminus_real_def
tff(fact_9831_times__real__def,axiom,
    times_times_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_zp) ).

% times_real_def
tff(fact_9832_plus__real__def,axiom,
    plus_plus_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_zn) ).

% plus_real_def
tff(fact_9833_Real_Opositive_Orsp,axiom,
    bNF_re2717234250851183977l_bool(realrel,fequal_bool,aTP_Lamm_zw,aTP_Lamm_zw) ).

% Real.positive.rsp
tff(fact_9834_cr__real__def,axiom,
    ! [X2: fun_nat_rat,Xa: real] :
      ( pp(aa_real_bool(aa_fun5258240734392065441l_bool(cr_real,X2),Xa))
    <=> ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X2),X2))
        & ( aa_fun_nat_rat_real(real2,X2) = Xa ) ) ) ).

% cr_real_def
tff(fact_9835_plus__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_zn,aTP_Lamm_zn) ).

% plus_real.rsp
tff(fact_9836_uminus__real_Orsp,axiom,
    pp(aa_fun1696419067750587730t_bool(aa_fun4901480949278917796t_bool(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_zl),aTP_Lamm_zl)) ).

% uminus_real.rsp
tff(fact_9837_times__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_zp,aTP_Lamm_zp) ).

% times_real.rsp
tff(fact_9838_gcd__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),gcd_gcd_int,gcd_gcd_int) ).

% gcd_integer.rsp
tff(fact_9839_inverse__real_Orsp,axiom,
    pp(aa_fun1696419067750587730t_bool(aa_fun4901480949278917796t_bool(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_zx),aTP_Lamm_zx)) ).

% inverse_real.rsp
tff(fact_9840_Real_Opositive_Otransfer,axiom,
    pp(aa_fun3352386287069783311l_bool(aa_fun7221085148252759141l_bool(bNF_re1458715823057739248l_bool(pcr_real,fequal_bool),aTP_Lamm_zw),positive2)) ).

% Real.positive.transfer
tff(fact_9841_inverse__rat_Otransfer,axiom,
    bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat,aTP_Lamm_zy,inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_9842_real_Orel__eq__transfer,axiom,
    bNF_re8935275315890131625l_bool(pcr_real,bNF_re1458715823057739248l_bool(pcr_real,fequal_bool),realrel,fequal_real) ).

% real.rel_eq_transfer
tff(fact_9843_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_9844_zero__real_Otransfer,axiom,
    pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,aTP_Lamm_zs),zero_zero_real)) ).

% zero_real.transfer
tff(fact_9845_one__real_Otransfer,axiom,
    pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,aTP_Lamm_zt),one_one_real)) ).

% one_real.transfer
tff(fact_9846_cr__real__eq,axiom,
    ! [X2: fun_nat_rat,Xa: real] :
      ( pp(aa_real_bool(aa_fun5258240734392065441l_bool(pcr_real,X2),Xa))
    <=> ( cauchy(X2)
        & ( aa_fun_nat_rat_real(real2,X2) = Xa ) ) ) ).

% cr_real_eq
tff(fact_9847_uminus__real_Otransfer,axiom,
    pp(aa_fun9137951878280015556l_bool(aa_fun9159459243913713430l_bool(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_zl),uminus_uminus_real)) ).

% uminus_real.transfer
tff(fact_9848_plus__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_zn,plus_plus_real) ).

% plus_real.transfer
tff(fact_9849_times__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_zp,times_times_real) ).

% times_real.transfer
tff(fact_9850_zero__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,product_Pair_int_int(zero_zero_int,one_one_int)),zero_zero_rat)) ).

% zero_rat.transfer
tff(fact_9851_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_zz,fract) ).

% Fract.transfer
tff(fact_9852_inverse__real_Otransfer,axiom,
    pp(aa_fun9137951878280015556l_bool(aa_fun9159459243913713430l_bool(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_zx),inverse_inverse_real)) ).

% inverse_real.transfer
tff(fact_9853_Rat_Opositive_Otransfer,axiom,
    bNF_re3535474257078034868l_bool(pcr_rat,fequal_bool,aTP_Lamm_wq,positive) ).

% Rat.positive.transfer
tff(fact_9854_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_wb),times_times_int) ).

% times_int.transfer
tff(fact_9855_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_wk),minus_minus_int) ).

% minus_int.transfer
tff(fact_9856_zero__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).

% zero_int.transfer
tff(fact_9857_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_aaa,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_9858_uminus__int_Otransfer,axiom,
    pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_wc)),uminus_uminus_int)) ).

% uminus_int.transfer
tff(fact_9859_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_9860_one__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).

% one_int.transfer
tff(fact_9861_less__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_we),ord_less_int) ).

% less_int.transfer
tff(fact_9862_less__eq__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wg),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_9863_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_wi),plus_plus_int) ).

% plus_int.transfer
tff(fact_9864_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_wb),produc4528391901023682385at_nat(aTP_Lamm_wb)) ).

% times_int.rsp
tff(fact_9865_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_9866_int_Orel__eq__transfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),intrel,fequal_int) ).

% int.rel_eq_transfer
tff(fact_9867_int_Obi__total,axiom,
    bi_tot896582865486249351at_int(pcr_int) ).

% int.bi_total
tff(fact_9868_zero__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).

% zero_int.rsp
tff(fact_9869_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( aa_Pro4240683299535347697at_int(abs_Integ,X) = aa_Pro4240683299535347697at_int(abs_Integ,Y) )
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,X),Y)) ) ).

% int.abs_eq_iff
tff(fact_9870_uminus__int_Orsp,axiom,
    pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_wc)),produc2626176000494625587at_nat(aTP_Lamm_wc))) ).

% uminus_int.rsp
tff(fact_9871_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_9872_one__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).

% one_int.rsp
tff(fact_9873_intrel__def,axiom,
    intrel = produc5023667602222594023t_bool(aTP_Lamm_aac) ).

% intrel_def
tff(fact_9874_less__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_we),produc5023667602222594023t_bool(aTP_Lamm_we)) ).

% less_int.rsp
tff(fact_9875_less__eq__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_wg),produc5023667602222594023t_bool(aTP_Lamm_wg)) ).

% less_eq_int.rsp
tff(fact_9876_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_wi),produc4528391901023682385at_nat(aTP_Lamm_wi)) ).

% plus_int.rsp
tff(fact_9877_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_wk),produc4528391901023682385at_nat(aTP_Lamm_wk)) ).

% minus_int.rsp
tff(fact_9878_le__enumerate,axiom,
    ! [S3: set_nat,N2: nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,S3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(infini8530281810654367211te_nat(S3),N2))) ) ).

% le_enumerate
tff(fact_9879_finite__le__enumerate,axiom,
    ! [S3: set_nat,N2: nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,S3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),aa_set_nat_nat(finite_card_nat,S3)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(infini8530281810654367211te_nat(S3),N2))) ) ) ).

% finite_le_enumerate
tff(fact_9880_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N2)),transi6264000038957366511cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% less_eq
tff(fact_9881_Least__eq__0,axiom,
    ! [P: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9882_Least__Suc2,axiom,
    ! [P: fun_nat_bool,N2: nat,Q: fun_nat_bool,M: nat] :
      ( pp(aa_nat_bool(P,N2))
     => ( pp(aa_nat_bool(Q,M))
       => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
         => ( ! [K2: nat] :
                ( pp(aa_nat_bool(P,aa_nat_nat(suc,K2)))
              <=> pp(aa_nat_bool(Q,K2)) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9883_Least__Suc,axiom,
    ! [P: fun_nat_bool,N2: nat] :
      ( pp(aa_nat_bool(P,N2))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_yj(P))) ) ) ) ).

% Least_Suc
tff(fact_9884_Sup__real__def,axiom,
    ! [X5: set_real] : comple1385675409528146559p_real(X5) = ord_Least_real(aTP_Lamm_aad(X5)) ).

% Sup_real_def
tff(fact_9885_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N2)),transi2905341329935302413cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% pred_nat_trancl_eq_le
tff(fact_9886_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,X)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,X)) ) ) ).

% of_real_sqrt
tff(fact_9887_inverse__rat_Orsp,axiom,
    bNF_re7145576690424134365nt_int(ratrel,ratrel,aTP_Lamm_zy,aTP_Lamm_zy) ).

% inverse_rat.rsp
tff(fact_9888_csqrt__0,axiom,
    aa_complex_complex(csqrt,zero_zero_complex) = zero_zero_complex ).

% csqrt_0
tff(fact_9889_csqrt__eq__0,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_complex(csqrt,Z2) = zero_zero_complex )
    <=> ( Z2 = zero_zero_complex ) ) ).

% csqrt_eq_0
tff(fact_9890_ratrel__iff,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),Y))
    <=> ( ( product_snd_int_int(X) != zero_zero_int )
        & ( product_snd_int_int(Y) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Y)),product_snd_int_int(X)) ) ) ) ).

% ratrel_iff
tff(fact_9891_zero__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,product_Pair_int_int(zero_zero_int,one_one_int)),product_Pair_int_int(zero_zero_int,one_one_int))) ).

% zero_rat.rsp
tff(fact_9892_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_zz,aTP_Lamm_zz) ).

% Fract.rsp
tff(fact_9893_ratrel__def,axiom,
    ! [X2: product_prod_int_int,Xa: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X2),Xa))
    <=> ( ( product_snd_int_int(X2) != 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(X2)),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(X2)) ) ) ) ).

% ratrel_def
tff(fact_9894_Rat_Opositive_Orsp,axiom,
    bNF_re1394333895916556125l_bool(ratrel,fequal_bool,aTP_Lamm_wq,aTP_Lamm_wq) ).

% Rat.positive.rsp
tff(fact_9895_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(X)),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(product_snd_int_int(X),product_fst_int_int(X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_9896_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( pp(aa_rat_bool(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,X)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_9897_Fract_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa_int_rat(aa_int_fun_int_rat(fract,Xa2),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(Xa2,X))) ).

% Fract.abs_eq
tff(fact_9898_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(zero_zero_int,one_one_int)) ).

% zero_rat_def
tff(fact_9899_inverse__rat__def,axiom,
    inverse_inverse_rat = map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat,aTP_Lamm_zy) ).

% inverse_rat_def
tff(fact_9900_csqrt_Ocode,axiom,
    ! [Z2: complex] : aa_complex_complex(csqrt,Z2) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(re,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z2)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z2)))),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,Z2)),aa_complex_real(re,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9901_complex__Im__fact,axiom,
    ! [N2: nat] : aa_complex_real(im,aa_nat_complex(semiri5044797733671781792omplex,N2)) = zero_zero_real ).

% complex_Im_fact
tff(fact_9902_complex__Im__of__int,axiom,
    ! [Z2: int] : aa_complex_real(im,aa_int_complex(ring_17405671764205052669omplex,Z2)) = zero_zero_real ).

% complex_Im_of_int
tff(fact_9903_complex__Im__of__nat,axiom,
    ! [N2: nat] : aa_complex_real(im,aa_nat_complex(semiri8010041392384452111omplex,N2)) = zero_zero_real ).

% complex_Im_of_nat
tff(fact_9904_Im__complex__of__real,axiom,
    ! [Z2: real] : aa_complex_real(im,aa_real_complex(real_V4546457046886955230omplex,Z2)) = zero_zero_real ).

% Im_complex_of_real
tff(fact_9905_Im__power__real,axiom,
    ! [X: complex,N2: nat] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(power_power_complex(X),N2)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_9906_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9907_Re__power__real,axiom,
    ! [X: complex,N2: nat] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(X),N2)) = aa_nat_real(power_power_real(aa_complex_real(re,X)),N2) ) ) ).

% Re_power_real
tff(fact_9908_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( aa_complex_real(re,Y) = zero_zero_real )
       => ( aa_real_real(cos_real,aa_complex_real(arg,Y)) = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_9909_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X)))
       => ( aa_complex_complex(csqrt,X) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_9910_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9911_zero__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(1)
tff(fact_9912_cmod__eq__Re,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_real(im,Z2) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z2) = aa_real_real(abs_abs_real,aa_complex_real(re,Z2)) ) ) ).

% cmod_eq_Re
tff(fact_9913_cmod__eq__Im,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_real(re,Z2) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z2) = aa_real_real(abs_abs_real,aa_complex_real(im,Z2)) ) ) ).

% cmod_eq_Im
tff(fact_9914_Im__eq__0,axiom,
    ! [Z2: complex] :
      ( ( aa_real_real(abs_abs_real,aa_complex_real(re,Z2)) = aa_complex_real(real_V1022390504157884413omplex,Z2) )
     => ( aa_complex_real(im,Z2) = zero_zero_real ) ) ).

% Im_eq_0
tff(fact_9915_complex__is__Int__iff,axiom,
    ! [Z2: complex] :
      ( pp(aa_set_complex_bool(member_complex(Z2),ring_1_Ints_complex))
    <=> ( ( aa_complex_real(im,Z2) = zero_zero_real )
        & ? [I3: int] : aa_complex_real(re,Z2) = aa_int_real(ring_1_of_int_real,I3) ) ) ).

% complex_is_Int_iff
tff(fact_9916_zero__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(2)
tff(fact_9917_complex__is__Nat__iff,axiom,
    ! [Z2: complex] :
      ( pp(aa_set_complex_bool(member_complex(Z2),semiri3842193898606819883omplex))
    <=> ( ( aa_complex_real(im,Z2) = zero_zero_real )
        & ? [I3: nat] : aa_complex_real(re,Z2) = aa_nat_real(semiri5074537144036343181t_real,I3) ) ) ).

% complex_is_Nat_iff
tff(fact_9918_csqrt__principal,axiom,
    ! [Z2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z2))))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z2)) = zero_zero_real )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z2)))) ) ) ).

% csqrt_principal
tff(fact_9919_Re__csqrt,axiom,
    ! [Z2: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z2)))) ).

% Re_csqrt
tff(fact_9920_complex__eq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9921_complex__neq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% complex_neq_0
tff(fact_9922_csqrt__square,axiom,
    ! [B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,B)))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B))) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9923_csqrt__unique,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z2 )
     => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,W2)))
          | ( ( aa_complex_real(re,W2) = zero_zero_real )
            & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W2))) ) )
       => ( aa_complex_complex(csqrt,Z2) = W2 ) ) ) ).

% csqrt_unique
tff(fact_9924_complex__unit__circle,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z2)),aa_complex_real(real_V1022390504157884413omplex,Z2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z2)),aa_complex_real(real_V1022390504157884413omplex,Z2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9925_cmod__plus__Re__le__0__iff,axiom,
    ! [Z2: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(re,Z2))),zero_zero_real))
    <=> ( aa_complex_real(re,Z2) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z2)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9926_csqrt_Osimps_I2_J,axiom,
    ! [Z2: complex] : aa_complex_real(im,aa_complex_complex(csqrt,Z2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z2)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z2)))),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,Z2)),aa_complex_real(re,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9927_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),zero_zero_real))
       => ( aa_complex_complex(csqrt,X) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9928_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,X)),zero_zero_real))
        | ( ( aa_complex_real(im,X) = zero_zero_real )
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X))) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,X)) ) ) ).

% csqrt_minus
tff(fact_9929_imaginary__unit_Osimps_I1_J,axiom,
    aa_complex_real(re,imaginary_unit) = zero_zero_real ).

% imaginary_unit.simps(1)
tff(fact_9930_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
tff(fact_9931_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_complex(complex2(X),Y) = imaginary_unit )
    <=> ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9932_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9933_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_9934_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_9935_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
       => ( ( ( K = zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
          & ( ( K != zero_zero_int )
           => ( aa_int_Code_integer(code_integer_of_int,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(divide_divide_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_9936_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa_set_complex_bool(member_complex(Y),real_V2521375963428798218omplex))
     => ( pp(aa_set_complex_bool(member_complex(X),real_V2521375963428798218omplex))
       => ( ( X = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
        <=> ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9937_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa_set_complex_bool(member_complex(Y),real_V2521375963428798218omplex))
     => ( pp(aa_set_complex_bool(member_complex(X),real_V2521375963428798218omplex))
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = X )
        <=> ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9938_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_9939_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9940_Complex__in__Reals,axiom,
    ! [X: real] : pp(aa_set_complex_bool(member_complex(aa_real_complex(complex2(X),zero_zero_real)),real_V2521375963428798218omplex)) ).

% Complex_in_Reals
tff(fact_9941_less__eq__integer__code_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_eq_integer_code(1)
tff(fact_9942_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_9943_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9944_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9945_gcd__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : gcd_gcd_Code_integer(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xa2),X)) ).

% gcd_integer.abs_eq
tff(fact_9946_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] : gcd_gcd_Code_integer(K,L) = abs_abs_Code_integer(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,L),zero_z3403309356797280102nteger),K,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9947_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9948_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9949_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9950_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X)) ) ).

% less_integer.abs_eq
tff(fact_9951_less__integer__code_I1_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_integer_code(1)
tff(fact_9952_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( abs_abs_Code_integer(K) = K ) ) ) ).

% abs_integer_code
tff(fact_9953_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
           => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).

% sgn_integer_code
tff(fact_9954_complex__is__Real__iff,axiom,
    ! [Z2: complex] :
      ( pp(aa_set_complex_bool(member_complex(Z2),real_V2521375963428798218omplex))
    <=> ( aa_complex_real(im,Z2) = zero_zero_real ) ) ).

% complex_is_Real_iff
tff(fact_9955_Suc_Orsp,axiom,
    pp(aa_fun_nat_nat_bool(aa_fun3185460345579526648t_bool(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc)) ).

% Suc.rsp
tff(fact_9956_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9957_less__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9958_less__eq__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_9959_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_9960_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_9961_less__integer_Orsp,axiom,
    bNF_re5780769757191362231t_bool(fequal_int,bNF_re9007465950226028919l_bool(fequal_int,fequal_bool),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9962_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9963_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( code_int_of_integer(K) = aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
       => ( ( ( K = zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = zero_zero_int ) )
          & ( ( K != zero_z3403309356797280102nteger )
           => ( code_int_of_integer(K) = produc1553301316500091796er_int(aTP_Lamm_aae,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_9964_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = one ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
       => ( code_num_of_integer(K) = produc7336495610019696514er_num(aTP_Lamm_aaf,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% num_of_integer_code
tff(fact_9965_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9966_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_iff
tff(fact_9967_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Xa2))) ) ).

% less_integer.rep_eq
tff(fact_9968_gcd__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(gcd_gcd_Code_integer(X,Xa2)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% gcd_integer.rep_eq
tff(fact_9969_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = zero_zero_nat ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
       => ( code_nat_of_integer(K) = produc1555791787009142072er_nat(aTP_Lamm_aag,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_9970_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_aah(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_9971_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,code_nat_of_integer(K)) = ord_max_Code_integer(zero_z3403309356797280102nteger,K) ).

% of_nat_of_integer
tff(fact_9972_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9973_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_9974_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9975_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,abs_abs_Code_integer(J)) ).

% divmod_abs_code(5)
tff(fact_9976_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_9977_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( ( L = zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
          & ( ( L != zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_aai(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_9978_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
               => ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_aaj(L),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( ( L = zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
              & ( ( L != zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_aak(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_9979_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_9980_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_aam,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9981_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_aao,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9982_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M7: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),M7),at_top_nat)
     => ( ! [M5: nat,N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M5))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,M5)),aa_nat_real(F,N))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_9983_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M7: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ce(F),M7),at_top_nat)
     => ( ! [M5: nat,N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),M5))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
             => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,M5))) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_9984_Bseq__mono__convergent,axiom,
    ! [X5: fun_nat_real] :
      ( bfun_nat_real(X5,at_top_nat)
     => ( ! [M5: nat,N: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M5),N))
           => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,M5)),aa_nat_real(X5,N))) )
       => topolo7531315842566124627t_real(X5) ) ) ).

% Bseq_mono_convergent
tff(fact_9985_convergent__realpow,axiom,
    ! [X: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
       => topolo7531315842566124627t_real(power_power_real(X)) ) ) ).

% convergent_realpow
tff(fact_9986_at__right__to__0,axiom,
    ! [A: real] : topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)) = filtermap_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aap,A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_right_to_0
tff(fact_9987_at__top__to__right,axiom,
    at_top_real = filtermap_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_top_to_right
tff(fact_9988_at__right__to__top,axiom,
    topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)) = filtermap_real_real(inverse_inverse_real,at_top_real) ).

% at_right_to_top
tff(fact_9989_filtermap__ln__at__right,axiom,
    filtermap_real_real(ln_ln_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) = at_bot_real ).

% filtermap_ln_at_right
tff(fact_9990_gcd__nat_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_vz),zero_zero_nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_9991_binomial__def,axiom,
    ! [N2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N2),K) = finite_card_set_nat(collect_set_nat(aa_nat8857836526276733660t_bool(aTP_Lamm_aaq(N2),K))) ).

% binomial_def
tff(fact_9992_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_am,aTP_Lamm_al),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_9993_set__encode__vimage__Suc,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_9994_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).

% vimage_Suc_insert_0
tff(fact_9995_vimage__Suc__insert__Suc,axiom,
    ! [N2: nat,A2: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,N2),A2)) = insert_nat(N2,vimage_nat_nat(suc,A2)) ).

% vimage_Suc_insert_Suc
tff(fact_9996_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,vimage_nat_nat(suc,F3)))
    <=> pp(aa_set_nat_bool(finite_finite_nat,F3)) ) ).

% finite_vimage_Suc_iff
tff(fact_9997_set__decode__div__2,axiom,
    ! [X: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,X)) ).

% set_decode_div_2
tff(fact_9998_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(sgn_sgn_int,K) ) ) ).

% division_segment_eq_sgn
tff(fact_9999_division__segment__nat__def,axiom,
    ! [N2: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,N2) = one_one_nat ).

% division_segment_nat_def
tff(fact_10000_division__segment__int__def,axiom,
    ! [K: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = one_one_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
       => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% division_segment_int_def
tff(fact_10001_less__eq__enat__def,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(M),N2))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aTP_Lamm_aar(M),fTrue),N2)) ) ).

% less_eq_enat_def
tff(fact_10002_less__enat__def,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M),N2))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aTP_Lamm_aas(N2),fFalse),M)) ) ).

% less_enat_def
tff(fact_10003_transp__realrel,axiom,
    transp_fun_nat_rat(realrel) ).

% transp_realrel
tff(fact_10004_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(product_Pair_int_int(zero_zero_int,A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_10005_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(product_Pair_int_int(A,zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_10006_of__rat__Real,axiom,
    ! [X: rat] : aa_rat_real(field_7254667332652039916t_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_zm(X)) ).

% of_rat_Real
tff(fact_10007_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
     => ? [Q2: rat] :
          ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,Q2)))
          & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_rat_real(field_7254667332652039916t_real,Q2)),Y)) ) ) ).

% of_rat_dense
tff(fact_10008_less__RealD,axiom,
    ! [Y5: fun_nat_rat,X: real] :
      ( cauchy(Y5)
     => ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_fun_nat_rat_real(real2,Y5)))
       => ? [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y5,N)))) ) ) ).

% less_RealD
tff(fact_10009_Real__leI,axiom,
    ! [X5: fun_nat_rat,Y: real] :
      ( cauchy(X5)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(X5,N))),Y))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X5)),Y)) ) ) ).

% Real_leI
tff(fact_10010_le__RealI,axiom,
    ! [Y5: fun_nat_rat,X: real] :
      ( cauchy(Y5)
     => ( ! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y5,N))))
       => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_fun_nat_rat_real(real2,Y5))) ) ) ).

% le_RealI
tff(fact_10011_compute__powr__real,axiom,
    ! [B: real,I: real] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_aat(B),I)) ) )
      & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
       => ( ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I )
           => ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = aa_nat_real(power_power_real(B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))) ) )
              & ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),I))
               => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(power_power_real(B),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I))))) ) ) ) )
          & ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) != I )
           => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B),I) = abort_real(literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fFalse,fTrue,fFalse,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,zero_zero_literal))))))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_aat(B),I)) ) ) ) ) ) ).

% compute_powr_real
tff(fact_10012_String_Oempty__neq__Literal,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,S: literal] : zero_zero_literal != literal2(B0,B1,B22,B32,B42,B52,B62,S) ).

% String.empty_neq_Literal
tff(fact_10013_inj__on__char__of__nat,axiom,
    pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))) ).

% inj_on_char_of_nat
tff(fact_10014_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_10015_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_10016_char_Osize_I2_J,axiom,
    ! [X15: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : size_size_char(char2(X15,X22,X32,X42,X52,X62,X72,X8)) = zero_zero_nat ).

% char.size(2)
tff(fact_10017_nat__of__char__less__256,axiom,
    ! [C: char] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% nat_of_char_less_256
tff(fact_10018_char_Osize__gen,axiom,
    ! [X15: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : size_char(char2(X15,X22,X32,X42,X52,X62,X72,X8)) = zero_zero_nat ).

% char.size_gen
tff(fact_10019_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N9: nat] :
        ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),M6))
         => ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N4))
             => pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N4),M6))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_10020_numeral__le__enat__iff,axiom,
    ! [M: num,N2: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),extended_enat2(N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),N2)) ) ).

% numeral_le_enat_iff
tff(fact_10021_elimnum,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N2))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimnum
tff(fact_10022_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),extended_enat2(N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2)) ) ).

% enat_ord_simps(2)
tff(fact_10023_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N2: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extended_enat2(M)),extended_enat2(N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2)) ) ).

% enat_ord_simps(1)
tff(fact_10024_plus__enat__simps_I1_J,axiom,
    ! [M: nat,N2: nat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(extended_enat2(M)),extended_enat2(N2)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N2)) ).

% plus_enat_simps(1)
tff(fact_10025_idiff__enat__0,axiom,
    ! [N2: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(extended_enat2(zero_zero_nat)),N2) = extended_enat2(zero_zero_nat) ).

% idiff_enat_0
tff(fact_10026_idiff__enat__0__right,axiom,
    ! [N2: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N2),extended_enat2(zero_zero_nat)) = N2 ).

% idiff_enat_0_right
tff(fact_10027_numeral__less__enat__iff,axiom,
    ! [M: num,N2: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),extended_enat2(N2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),N2)) ) ).

% numeral_less_enat_iff
tff(fact_10028_enat__iless,axiom,
    ! [N2: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N2),extended_enat2(M)))
     => ? [K2: nat] : N2 = extended_enat2(K2) ) ).

% enat_iless
tff(fact_10029_less__enatE,axiom,
    ! [N2: extended_enat,M: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N2),extended_enat2(M)))
     => ~ ! [K2: nat] :
            ( ( N2 = extended_enat2(K2) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),M)) ) ) ).

% less_enatE
tff(fact_10030_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: bool,B: bool,Uu: extended_enat] : vEBT_VEBT_elim_dead(vEBT_Leaf(A,B),Uu) = vEBT_Leaf(A,B) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_10031_enat__1__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( one_on7984719198319812577d_enat = extended_enat2(X) )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(2)
tff(fact_10032_enat__1__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = one_on7984719198319812577d_enat )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(1)
tff(fact_10033_one__enat__def,axiom,
    one_on7984719198319812577d_enat = extended_enat2(one_one_nat) ).

% one_enat_def
tff(fact_10034_Suc__ile__eq,axiom,
    ! [M: nat,N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extended_enat2(aa_nat_nat(suc,M))),N2))
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),N2)) ) ).

% Suc_ile_eq
tff(fact_10035_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat = extended_enat2(X) )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_10036_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = zero_z5237406670263579293d_enat )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_10037_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = extended_enat2(zero_zero_nat) ).

% zero_enat_def
tff(fact_10038_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extended_enat2(L)) = vEBT_Node(Info2,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_aau(Deg),TreeList2)),vEBT_VEBT_elim_dead(Summary,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_10039_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N2: nat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(X),Y)),extended_enat2(N2)))
    <=> ? [Y7: nat,X9: nat] :
          ( ( X = extended_enat2(X9) )
          & ( Y = extended_enat2(Y7) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X9),Y7)),N2)) ) ) ).

% iadd_le_enat_iff
tff(fact_10040_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa2) = Y )
     => ( ! [A3: bool,B3: bool] :
            ( ( X = vEBT_Leaf(A3,B3) )
           => ( Y != vEBT_Leaf(A3,B3) ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
             => ( ( Xa2 = extend5688581933313929465d_enat )
               => ( Y != vEBT_Node(Info,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_aau(Deg2),TreeList),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ! [L3: nat] :
                    ( ( Xa2 = extended_enat2(L3) )
                   => ( Y != vEBT_Node(Info,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_aau(Deg2),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_10041_elimcomplete,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary)),N2))
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimcomplete
tff(fact_10042_enat__ord__simps_I4_J,axiom,
    ! [Q4: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Q4),extend5688581933313929465d_enat))
    <=> ( Q4 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
tff(fact_10043_enat__ord__simps_I6_J,axiom,
    ! [Q4: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),Q4)) ).

% enat_ord_simps(6)
tff(fact_10044_idiff__self,axiom,
    ! [N2: extended_enat] :
      ( ( N2 != extend5688581933313929465d_enat )
     => ( aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(N2),N2) = zero_z5237406670263579293d_enat ) ) ).

% idiff_self
tff(fact_10045_idiff__infinity__right,axiom,
    ! [A: nat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(extended_enat2(A)),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ).

% idiff_infinity_right
tff(fact_10046_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extended_enat2(M)),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extended_enat2(M)),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
tff(fact_10047_times__enat__simps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extended_enat2(N2)) = zero_z5237406670263579293d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extended_enat2(N2)) = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
tff(fact_10048_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: produc7272778201969148633d_enat] :
      ( ! [A3: bool,B3: bool,Uu2: extended_enat] : X != produc581526299967858633d_enat(vEBT_Leaf(A3,B3),Uu2)
     => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X != produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extend5688581933313929465d_enat)
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,L3: nat] : X != produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extended_enat2(L3)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_10049_enat__ord__code_I4_J,axiom,
    ! [M: nat] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),extend5688581933313929465d_enat)) ).

% enat_ord_code(4)
tff(fact_10050_less__infinityE,axiom,
    ! [N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N2),extend5688581933313929465d_enat))
     => ~ ! [K2: nat] : N2 != extended_enat2(K2) ) ).

% less_infinityE
tff(fact_10051_infinity__ilessE,axiom,
    ! [M: nat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),extended_enat2(M))) ).

% infinity_ilessE
tff(fact_10052_Sup__enat__def,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ( A2 = bot_bo7653980558646680370d_enat )
       => ( comple4398354569131411667d_enat(A2) = zero_z5237406670263579293d_enat ) )
      & ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ( pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
           => ( comple4398354569131411667d_enat(A2) = lattic921264341876707157d_enat(A2) ) )
          & ( ~ pp(aa_set3347977610044113549t_bool(finite4001608067531595151d_enat,A2))
           => ( comple4398354569131411667d_enat(A2) = extend5688581933313929465d_enat ) ) ) ) ) ).

% Sup_enat_def
tff(fact_10053_infinity__ne__i0,axiom,
    extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).

% infinity_ne_i0
tff(fact_10054_imult__is__infinity,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) = extend5688581933313929465d_enat )
    <=> ( ( ( A = extend5688581933313929465d_enat )
          & ( B != zero_z5237406670263579293d_enat ) )
        | ( ( B = extend5688581933313929465d_enat )
          & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).

% imult_is_infinity
tff(fact_10055_enat__add__left__cancel__less,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),B)),aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(A),C)))
    <=> ( ( A != extend5688581933313929465d_enat )
        & pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),C)) ) ) ).

% enat_add_left_cancel_less
tff(fact_10056_imult__infinity,axiom,
    ! [N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N2))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),N2) = extend5688581933313929465d_enat ) ) ).

% imult_infinity
tff(fact_10057_imult__infinity__right,axiom,
    ! [N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N2))
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(N2),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
tff(fact_10058_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend5688581933313929465d_enat) = vEBT_Node(Info2,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_aau(Deg),TreeList2),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_10059_times__enat__def,axiom,
    ! [M: extended_enat,N2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(M),N2) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_aaw(N2),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(N2),zero_z5237406670263579293d_enat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),M) ).

% times_enat_def
tff(fact_10060_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa2) = Y )
     => ( accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(X,Xa2))
       => ( ! [A3: bool,B3: bool] :
              ( ( X = vEBT_Leaf(A3,B3) )
             => ( ( Y = vEBT_Leaf(A3,B3) )
               => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Leaf(A3,B3),Xa2)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ( ( Xa2 = extend5688581933313929465d_enat )
                 => ( ( Y = vEBT_Node(Info,Deg2,map_VE8901447254227204932T_VEBT(aTP_Lamm_aau(Deg2),TreeList),vEBT_VEBT_elim_dead(Summary2,extend5688581933313929465d_enat)) )
                   => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
                 => ! [L3: nat] :
                      ( ( Xa2 = extended_enat2(L3) )
                     => ( ( Y = vEBT_Node(Info,Deg2,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_aau(Deg2),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
                       => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info,Deg2,TreeList,Summary2),extended_enat2(L3))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_10061_plus__enat__def,axiom,
    ! [M: extended_enat,N2: extended_enat] : aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(M),N2) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_aay(N2),extend5688581933313929465d_enat),M) ).

% plus_enat_def
tff(fact_10062_diff__enat__def,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(minus_3235023915231533773d_enat(A),B) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_aba(B),extend5688581933313929465d_enat),A) ).

% diff_enat_def
tff(fact_10063_eSuc__def,axiom,
    ! [I: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,I) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_abb,extend5688581933313929465d_enat),I) ).

% eSuc_def
tff(fact_10064_iless__Suc__eq,axiom,
    ! [M: nat,N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(extended_enat2(M)),aa_Ext7969882492699152700d_enat(extended_eSuc,N2)))
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(extended_enat2(M)),N2)) ) ).

% iless_Suc_eq
tff(fact_10065_eSuc__mono,axiom,
    ! [N2: extended_enat,M: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,N2)),aa_Ext7969882492699152700d_enat(extended_eSuc,M)))
    <=> pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N2),M)) ) ).

% eSuc_mono
tff(fact_10066_iless__eSuc0,axiom,
    ! [N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N2),aa_Ext7969882492699152700d_enat(extended_eSuc,zero_z5237406670263579293d_enat)))
    <=> ( N2 = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
tff(fact_10067_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( extended_enat2(Y) = aa_Ext7969882492699152700d_enat(extended_eSuc,X) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( extended_enat2(N4) = X ) ) ) ).

% enat_eSuc_iff
tff(fact_10068_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( aa_Ext7969882492699152700d_enat(extended_eSuc,X) = extended_enat2(Y) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( X = extended_enat2(N4) ) ) ) ).

% eSuc_enat_iff
tff(fact_10069_eSuc__enat,axiom,
    ! [N2: nat] : aa_Ext7969882492699152700d_enat(extended_eSuc,extended_enat2(N2)) = extended_enat2(aa_nat_nat(suc,N2)) ).

% eSuc_enat
tff(fact_10070_i0__iless__eSuc,axiom,
    ! [N2: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(extended_eSuc,N2))) ).

% i0_iless_eSuc
tff(fact_10071_one__eSuc,axiom,
    one_on7984719198319812577d_enat = aa_Ext7969882492699152700d_enat(extended_eSuc,zero_z5237406670263579293d_enat) ).

% one_eSuc
tff(fact_10072_zero__ne__eSuc,axiom,
    ! [N2: extended_enat] : zero_z5237406670263579293d_enat != aa_Ext7969882492699152700d_enat(extended_eSuc,N2) ).

% zero_ne_eSuc
tff(fact_10073_not__eSuc__ilei0,axiom,
    ! [N2: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,N2)),zero_z5237406670263579293d_enat)) ).

% not_eSuc_ilei0
tff(fact_10074_ileI1,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M),N2))
     => pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,M)),N2)) ) ).

% ileI1
tff(fact_10075_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),less_than))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_than_iff
tff(fact_10076_pair__less__def,axiom,
    fun_pair_less = lex_prod_nat_nat(less_than,less_than) ).

% pair_less_def
tff(fact_10077_Quotient__real,axiom,
    quotie6870213296501934825t_real(realrel,real2,rep_real,cr_real) ).

% Quotient_real
tff(fact_10078_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) )
      & ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
      <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ) ).

% complex_div_gt_0
tff(fact_10079_complex__cnj__zero,axiom,
    aa_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).

% complex_cnj_zero
tff(fact_10080_complex__cnj__zero__iff,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_complex(cnj,Z2) = zero_zero_complex )
    <=> ( Z2 = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
tff(fact_10081_complex__In__mult__cnj__zero,axiom,
    ! [Z2: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_10082_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_10083_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_10084_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Re_complex_div_gt_0
tff(fact_10085_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Re_complex_div_lt_0
tff(fact_10086_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Re_complex_div_ge_0
tff(fact_10087_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Re_complex_div_le_0
tff(fact_10088_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Im_complex_div_gt_0
tff(fact_10089_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Im_complex_div_lt_0
tff(fact_10090_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).

% Im_complex_div_ge_0
tff(fact_10091_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).

% Im_complex_div_le_0
tff(fact_10092_cr__int__def,axiom,
    ! [X2: product_prod_nat_nat] : aa_Pro2928840338864870335t_bool(cr_int,X2) = aa_int_fun_int_bool(fequal_int,aa_Pro4240683299535347697at_int(abs_Integ,X2)) ).

% cr_int_def
tff(fact_10093_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_10094_Quotient__int,axiom,
    quotie1194848508323700631at_int(intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_10095_gcd__nat_Osemilattice__neutr__axioms,axiom,
    pp(aa_nat_bool(semila9081495762789891438tr_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_10096_max__nat_Osemilattice__neutr__axioms,axiom,
    pp(aa_nat_bool(semila9081495762789891438tr_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_10097_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_10098_Restr__natLeq,axiom,
    ! [N2: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,N2)),aTP_Lamm_abc(N2))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abd(N2))) ).

% Restr_natLeq
tff(fact_10099_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_10100_Restr__natLeq2,axiom,
    ! [N2: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N2),aTP_Lamm_abe(N2))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_abd(N2))) ).

% Restr_natLeq2
tff(fact_10101_natLeq__underS__less,axiom,
    ! [N2: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N2) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,N2)) ).

% natLeq_underS_less
tff(fact_10102_wf__less,axiom,
    pp(aa_set3135850010578026034t_bool(wf_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)))) ).

% wf_less
tff(fact_10103_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat(fun_pair_less) ).

% wf_pair_less
tff(fact_10104_wf__int__ge__less__than2,axiom,
    ! [D: int] : wf_int(int_ge_less_than2(D)) ).

% wf_int_ge_less_than2
tff(fact_10105_wf__int__ge__less__than,axiom,
    ! [D: int] : wf_int(int_ge_less_than(D)) ).

% wf_int_ge_less_than
tff(fact_10106_pair__leq__def,axiom,
    fun_pair_leq = sup_su718114333110466843at_nat(fun_pair_less,id_Pro2258643101195443293at_nat) ).

% pair_leq_def
tff(fact_10107_max__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(ord_max_nat),zero_zero_nat)) ).

% max_nat.monoid_axioms
tff(fact_10108_gcd__nat_Omonoid__axioms,axiom,
    pp(aa_nat_bool(monoid_nat(gcd_gcd_nat),zero_zero_nat)) ).

% gcd_nat.monoid_axioms
tff(fact_10109_rcis__eq__zero__iff,axiom,
    ! [R2: real,A: real] :
      ( ( aa_real_complex(rcis(R2),A) = zero_zero_complex )
    <=> ( R2 = zero_zero_real ) ) ).

% rcis_eq_zero_iff
tff(fact_10110_rcis__zero__mod,axiom,
    ! [A: real] : aa_real_complex(rcis(zero_zero_real),A) = zero_zero_complex ).

% rcis_zero_mod
tff(fact_10111_rcis__zero__arg,axiom,
    ! [R2: real] : aa_real_complex(rcis(R2),zero_zero_real) = aa_real_complex(real_V4546457046886955230omplex,R2) ).

% rcis_zero_arg
tff(fact_10112_Lcm__eq__0__I__nat,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
     => ( aa_set_nat_nat(gcd_Lcm_nat,A2) = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
tff(fact_10113_abs__Lcm__eq,axiom,
    ! [K5: set_int] : aa_int_int(abs_abs_int,gcd_Lcm_int(K5)) = gcd_Lcm_int(K5) ).

% abs_Lcm_eq
tff(fact_10114_Lcm__0__iff__nat,axiom,
    ! [A2: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,A2))
     => ( ( aa_set_nat_nat(gcd_Lcm_nat,A2) = zero_zero_nat )
      <=> pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).

% Lcm_0_iff_nat
tff(fact_10115_Lcm__abs__eq,axiom,
    ! [K5: set_int] : gcd_Lcm_int(image_int_int(abs_abs_int,K5)) = gcd_Lcm_int(K5) ).

% Lcm_abs_eq
tff(fact_10116_Lcm__int__eq,axiom,
    ! [N5: set_nat] : gcd_Lcm_int(image_nat_int(semiri1314217659103216013at_int,N5)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,N5)) ).

% Lcm_int_eq
tff(fact_10117_Lcm__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(aTP_Lamm_wn,K5)) = aa_int_nat(nat2,gcd_Lcm_int(K5)) ).

% Lcm_nat_abs_eq
tff(fact_10118_Gcd__nat__def,axiom,
    ! [M7: set_nat] : aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(gcd_Lcm_nat,collect_nat(aTP_Lamm_abf(M7))) ).

% Gcd_nat_def
tff(fact_10119_dvd__Lcm__nat,axiom,
    ! [M: nat,M7: set_nat] :
      ( pp(aa_set_nat_bool(member_nat(M),M7))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_set_nat_nat(gcd_Lcm_nat,M7))) ) ).

% dvd_Lcm_nat
tff(fact_10120_Lcm__dvd__nat,axiom,
    ! [M7: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( pp(aa_set_nat_bool(member_nat(X4),M7))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X4),N2)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_set_nat_nat(gcd_Lcm_nat,M7)),N2)) ) ).

% Lcm_dvd_nat
tff(fact_10121_Lcm__least__int,axiom,
    ! [A2: set_int,A: int] :
      ( ! [B3: int] :
          ( pp(aa_set_int_bool(member_int(B3),A2))
         => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B3),A)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,gcd_Lcm_int(A2)),A)) ) ).

% Lcm_least_int
tff(fact_10122_dvd__Lcm__int,axiom,
    ! [M: int,M7: set_int] :
      ( pp(aa_set_int_bool(member_int(M),M7))
     => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),gcd_Lcm_int(M7))) ) ).

% dvd_Lcm_int
tff(fact_10123_Lcm__nat__infinite,axiom,
    ! [M7: set_nat] :
      ( ~ pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( aa_set_nat_nat(gcd_Lcm_nat,M7) = zero_zero_nat ) ) ).

% Lcm_nat_infinite
tff(fact_10124_Lcm__nat__empty,axiom,
    aa_set_nat_nat(gcd_Lcm_nat,bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_10125_Lcm__int__greater__eq__0,axiom,
    ! [K5: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),gcd_Lcm_int(K5))) ).

% Lcm_int_greater_eq_0
tff(fact_10126_Lcm__int__def,axiom,
    ! [K5: set_int] : gcd_Lcm_int(K5) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),K5))) ).

% Lcm_int_def
tff(fact_10127_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Lcm_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_lcm_int,Xs),one_one_int) ).

% Lcm_int_set_eq_fold
tff(fact_10128_Lcm__eq__Max__nat,axiom,
    ! [M7: set_nat] :
      ( pp(aa_set_nat_bool(finite_finite_nat,M7))
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M7))
         => ( ! [M5: nat,N: nat] :
                ( pp(aa_set_nat_bool(member_nat(M5),M7))
               => ( pp(aa_set_nat_bool(member_nat(N),M7))
                 => pp(aa_set_nat_bool(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M5),N)),M7)) ) )
           => ( aa_set_nat_nat(gcd_Lcm_nat,M7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,M7) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_10129_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_zy,Uu) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(Uu)),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(product_snd_int_int(Uu),product_fst_int_int(Uu))) ).

% ATP.lambda_1
tff(fact_10130_ATP_Olambda__2,axiom,
    ! [Uu: fun_nat_rat] : aa_fun765758790203053448at_rat(aTP_Lamm_zx,Uu) = if_fun_nat_rat(vanishes(Uu),aTP_Lamm_zs,aTP_Lamm_zq(Uu)) ).

% ATP.lambda_2
tff(fact_10131_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ca,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_3
tff(fact_10132_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_iu,Uu) = aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_4
tff(fact_10133_ATP_Olambda__5,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_vk,Uu))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_5
tff(fact_10134_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_lo,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(power_power_real(zero_zero_real),Uu)) ).

% ATP.lambda_6
tff(fact_10135_ATP_Olambda__7,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_xq,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_7
tff(fact_10136_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ys,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_8
tff(fact_10137_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_zj,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_9
tff(fact_10138_ATP_Olambda__10,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ly,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_10
tff(fact_10139_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_lz,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_11
tff(fact_10140_ATP_Olambda__12,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ma,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_12
tff(fact_10141_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_mb,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_13
tff(fact_10142_ATP_Olambda__14,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_mc,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_14
tff(fact_10143_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_aaa,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_15
tff(fact_10144_ATP_Olambda__16,axiom,
    ! [Uu: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_wq,Uu))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu)))) ) ).

% ATP.lambda_16
tff(fact_10145_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_xw,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_17
tff(fact_10146_ATP_Olambda__18,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_xo,Uu) = suminf_real(aTP_Lamm_bz(Uu)) ).

% ATP.lambda_18
tff(fact_10147_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_wu,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_gy,Uu)) ).

% ATP.lambda_19
tff(fact_10148_ATP_Olambda__20,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_aao,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_aan(Uu)))) ).

% ATP.lambda_20
tff(fact_10149_ATP_Olambda__21,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_aam,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_aal(Uu)))) ).

% ATP.lambda_21
tff(fact_10150_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_xy,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_22
tff(fact_10151_ATP_Olambda__23,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_wo,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_23
tff(fact_10152_ATP_Olambda__24,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_vd,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_24
tff(fact_10153_ATP_Olambda__25,axiom,
    ! [Uu: nat] : aa_nat_Extended_enat(aTP_Lamm_abb,Uu) = extended_enat2(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_25
tff(fact_10154_ATP_Olambda__26,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_wn,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_26
tff(fact_10155_ATP_Olambda__27,axiom,
    ! [Uu: fun_nat_rat] :
      ( pp(aa_fun_nat_rat_bool(aTP_Lamm_zw,Uu))
    <=> ? [R5: rat] :
          ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
          & ? [K3: nat] :
            ! [N4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N4))
             => pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu,N4))) ) ) ) ).

% ATP.lambda_27
tff(fact_10156_ATP_Olambda__28,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_yw,Uu))
    <=> ? [I3: int,N4: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_28
tff(fact_10157_ATP_Olambda__29,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_yx,Uu))
    <=> ? [I3: int,J3: int] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_29
tff(fact_10158_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pi(Uu),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_30
tff(fact_10159_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pl,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_31
tff(fact_10160_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pj(Uu),Uua) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_rat) ).

% ATP.lambda_32
tff(fact_10161_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_pk,Uu),Uua) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_33
tff(fact_10162_ATP_Olambda__34,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ht(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),zero_zero_real),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_34
tff(fact_10163_ATP_Olambda__35,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_aae,Uu),Uua) = aa_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),one_one_int)) ).

% ATP.lambda_35
tff(fact_10164_ATP_Olambda__36,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_aaf,Uu),Uua) = aa_num_num(if_num(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger),aa_num_num(plus_plus_num(code_num_of_integer(Uu)),code_num_of_integer(Uu))),aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(code_num_of_integer(Uu)),code_num_of_integer(Uu))),one)) ).

% ATP.lambda_36
tff(fact_10165_ATP_Olambda__37,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_aag,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one_nat)) ).

% ATP.lambda_37
tff(fact_10166_ATP_Olambda__38,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_vl,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_38
tff(fact_10167_ATP_Olambda__39,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_zz,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int),product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(Uu,Uua)) ).

% ATP.lambda_39
tff(fact_10168_ATP_Olambda__40,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aaw(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_aav(Uua),aa_Ext7969882492699152700d_enat(if_Extended_enat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),zero_z5237406670263579293d_enat),extend5688581933313929465d_enat)),Uu) ).

% ATP.lambda_40
tff(fact_10169_ATP_Olambda__41,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aba(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_aaz(Uua),zero_z5237406670263579293d_enat),Uu) ).

% ATP.lambda_41
tff(fact_10170_ATP_Olambda__42,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aay(Uu),Uua) = aa_Ext7969882492699152700d_enat(extend3600170679010898289d_enat(aTP_Lamm_aax(Uua),extend5688581933313929465d_enat),Uu) ).

% ATP.lambda_42
tff(fact_10171_ATP_Olambda__43,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_aar(Uu),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(aTP_Lamm_am,Uua),fFalse),Uu)) ) ).

% ATP.lambda_43
tff(fact_10172_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pm(Uu),Uua) = aa_complex_complex(if_complex(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_complex) ).

% ATP.lambda_44
tff(fact_10173_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_real) ).

% ATP.lambda_45
tff(fact_10174_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pn(Uu),Uua) = aa_rat_rat(if_rat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_rat) ).

% ATP.lambda_46
tff(fact_10175_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_po,Uu),Uua) = aa_int_int(if_int(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))),zero_zero_int) ).

% ATP.lambda_47
tff(fact_10176_ATP_Olambda__48,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_aas(Uu),Uua))
    <=> pp(aa_Ext2772979874139974381t_bool(extend8126323780198025762t_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),fTrue),Uu)) ) ).

% ATP.lambda_48
tff(fact_10177_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_br(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_49
tff(fact_10178_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_jr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_50
tff(fact_10179_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_sv(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_51
tff(fact_10180_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_sw(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_52
tff(fact_10181_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_53
tff(fact_10182_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_sx(Uu),Uua) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_54
tff(fact_10183_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sy,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_55
tff(fact_10184_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ta(Uu),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_56
tff(fact_10185_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_real(power_power_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_57
tff(fact_10186_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_tb(Uu),Uua) = aa_rat_rat(divide_divide_rat(aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_58
tff(fact_10187_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_vi,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_59
tff(fact_10188_ATP_Olambda__60,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_zb(Uu),Uua))
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi)) ) ) ).

% ATP.lambda_60
tff(fact_10189_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_vv(Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_61
tff(fact_10190_ATP_Olambda__62,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_vu(Uu),Uua))
    <=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).

% ATP.lambda_62
tff(fact_10191_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_bz(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_63
tff(fact_10192_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xp(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_64
tff(fact_10193_ATP_Olambda__65,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_65
tff(fact_10194_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_tp(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_66
tff(fact_10195_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_tq(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_67
tff(fact_10196_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_68
tff(fact_10197_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_tr(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_69
tff(fact_10198_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ts,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_70
tff(fact_10199_ATP_Olambda__71,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_zc(Uu),Uua))
    <=> ( ( Uu = aa_int_rat(aa_int_fun_int_rat(fract,product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(Uua)))
        & pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua))) ) ) ).

% ATP.lambda_71
tff(fact_10200_ATP_Olambda__72,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_qq(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_72
tff(fact_10201_ATP_Olambda__73,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_qs(Uu),Uua) = aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_73
tff(fact_10202_ATP_Olambda__74,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qr(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_74
tff(fact_10203_ATP_Olambda__75,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_md(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_75
tff(fact_10204_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_mf(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_76
tff(fact_10205_ATP_Olambda__77,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_me(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_77
tff(fact_10206_ATP_Olambda__78,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ri(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)) ).

% ATP.lambda_78
tff(fact_10207_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_rk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(Uu),Uua)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_79
tff(fact_10208_ATP_Olambda__80,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rj(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)) ).

% ATP.lambda_80
tff(fact_10209_ATP_Olambda__81,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_vz,Uu),Uua))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_81
tff(fact_10210_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_lp(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_82
tff(fact_10211_ATP_Olambda__83,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yt,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_83
tff(fact_10212_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_84
tff(fact_10213_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_85
tff(fact_10214_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_86
tff(fact_10215_ATP_Olambda__87,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_bq(Uu),Uua))
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_87
tff(fact_10216_ATP_Olambda__88,axiom,
    ! [Uu: nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_bp(Uu),Uua))
    <=> ( aa_nat_real(power_power_real(Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_88
tff(fact_10217_ATP_Olambda__89,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_bu,Uu),Uua))
    <=> ( member_real(Uua,ring_1_Ints_real)
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Uua)),Uu)) ) ) ).

% ATP.lambda_89
tff(fact_10218_ATP_Olambda__90,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_bv,Uu),Uua))
    <=> ( member_rat(Uua,ring_1_Ints_rat)
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Uua)),Uu)) ) ) ).

% ATP.lambda_90
tff(fact_10219_ATP_Olambda__91,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_91
tff(fact_10220_ATP_Olambda__92,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_by(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_92
tff(fact_10221_ATP_Olambda__93,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_vo,Uu),Uua))
    <=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_93
tff(fact_10222_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ox(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_94
tff(fact_10223_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oy(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_95
tff(fact_10224_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_oz(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_96
tff(fact_10225_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pa(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_97
tff(fact_10226_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iz(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_98
tff(fact_10227_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_iw(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_99
tff(fact_10228_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_iy(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_100
tff(fact_10229_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ix(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_101
tff(fact_10230_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_id(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_102
tff(fact_10231_ATP_Olambda__103,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ib(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_103
tff(fact_10232_ATP_Olambda__104,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ic(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_104
tff(fact_10233_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_bs(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_105
tff(fact_10234_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_bt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(power_power_real(zero_zero_real),Uua)) ).

% ATP.lambda_106
tff(fact_10235_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_fz(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_107
tff(fact_10236_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_cx(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_108
tff(fact_10237_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_it(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_109
tff(fact_10238_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ir(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_110
tff(fact_10239_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_is(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_111
tff(fact_10240_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iq(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_112
tff(fact_10241_ATP_Olambda__113,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_io(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_113
tff(fact_10242_ATP_Olambda__114,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ip(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_114
tff(fact_10243_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_km(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_115
tff(fact_10244_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kk(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_116
tff(fact_10245_ATP_Olambda__117,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kl(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_117
tff(fact_10246_ATP_Olambda__118,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qc(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_118
tff(fact_10247_ATP_Olambda__119,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qe(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_119
tff(fact_10248_ATP_Olambda__120,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pz(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_120
tff(fact_10249_ATP_Olambda__121,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_pw(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uua)),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_121
tff(fact_10250_ATP_Olambda__122,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_uu(Uu),Uua) = aa_real_complex(complex2(aa_complex_real(Uu,Uua)),zero_zero_real) ).

% ATP.lambda_122
tff(fact_10251_ATP_Olambda__123,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_ut(Uu),Uua) = aa_real_complex(complex2(aa_nat_real(Uu,Uua)),zero_zero_real) ).

% ATP.lambda_123
tff(fact_10252_ATP_Olambda__124,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_hd(Uu),Uua))
    <=> ( aa_Ext2541572767299617110t_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_124
tff(fact_10253_ATP_Olambda__125,axiom,
    ! [Uu: fun_Ex4152069839811072081at_rat,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_hh(Uu),Uua))
    <=> ( aa_Extended_enat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_125
tff(fact_10254_ATP_Olambda__126,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_hc(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_126
tff(fact_10255_ATP_Olambda__127,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_hg(Uu),Uua))
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_127
tff(fact_10256_ATP_Olambda__128,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ha(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_128
tff(fact_10257_ATP_Olambda__129,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_he(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_129
tff(fact_10258_ATP_Olambda__130,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_hi(Uu),Uua))
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_130
tff(fact_10259_ATP_Olambda__131,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_hb(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_131
tff(fact_10260_ATP_Olambda__132,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_hf(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_132
tff(fact_10261_ATP_Olambda__133,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_hj(Uu),Uua))
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_133
tff(fact_10262_ATP_Olambda__134,axiom,
    ! [Uu: fun_Ex3741802288643809319omplex,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_nw(Uu),Uua))
    <=> ( aa_Ext8748583412541189208omplex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_134
tff(fact_10263_ATP_Olambda__135,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aTP_Lamm_oa(Uu),Uua))
    <=> ( aa_Ext2541572767299617110t_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_135
tff(fact_10264_ATP_Olambda__136,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_nv(Uu),Uua))
    <=> ( aa_complex_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_136
tff(fact_10265_ATP_Olambda__137,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_nz(Uu),Uua))
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_137
tff(fact_10266_ATP_Olambda__138,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_nt(Uu),Uua))
    <=> ( aa_real_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_138
tff(fact_10267_ATP_Olambda__139,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_nx(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_139
tff(fact_10268_ATP_Olambda__140,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_ob(Uu),Uua))
    <=> ( aa_real_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_140
tff(fact_10269_ATP_Olambda__141,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_nu(Uu),Uua))
    <=> ( aa_int_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_141
tff(fact_10270_ATP_Olambda__142,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ny(Uu),Uua))
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_142
tff(fact_10271_ATP_Olambda__143,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_oc(Uu),Uua))
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_143
tff(fact_10272_ATP_Olambda__144,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xs(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_144
tff(fact_10273_ATP_Olambda__145,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yg(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_xr(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_145
tff(fact_10274_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ve,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_146
tff(fact_10275_ATP_Olambda__147,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_va(Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_147
tff(fact_10276_ATP_Olambda__148,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_uz(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_148
tff(fact_10277_ATP_Olambda__149,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ev(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).

% ATP.lambda_149
tff(fact_10278_ATP_Olambda__150,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_eu(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_150
tff(fact_10279_ATP_Olambda__151,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_wm(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_151
tff(fact_10280_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_wz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(power_power_real(aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_152
tff(fact_10281_ATP_Olambda__153,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_ln(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_153
tff(fact_10282_ATP_Olambda__154,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_lm(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_154
tff(fact_10283_ATP_Olambda__155,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_pf(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_155
tff(fact_10284_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ph,Uu),Uua) = aa_nat_real(gbinomial_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_156
tff(fact_10285_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pg(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_157
tff(fact_10286_ATP_Olambda__158,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lq(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_158
tff(fact_10287_ATP_Olambda__159,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_lr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_159
tff(fact_10288_ATP_Olambda__160,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ls(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_160
tff(fact_10289_ATP_Olambda__161,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_to,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_161
tff(fact_10290_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yd(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_162
tff(fact_10291_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_aau(Uu),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% ATP.lambda_163
tff(fact_10292_ATP_Olambda__164,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_vf,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).

% ATP.lambda_164
tff(fact_10293_ATP_Olambda__165,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yc(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_165
tff(fact_10294_ATP_Olambda__166,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xz(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_166
tff(fact_10295_ATP_Olambda__167,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_wx(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_167
tff(fact_10296_ATP_Olambda__168,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_oh(Uu),Uua) = aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_168
tff(fact_10297_ATP_Olambda__169,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_og(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_169
tff(fact_10298_ATP_Olambda__170,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_of(Uu),Uua) = aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_170
tff(fact_10299_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_171
tff(fact_10300_ATP_Olambda__172,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_oj(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_172
tff(fact_10301_ATP_Olambda__173,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_on(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_173
tff(fact_10302_ATP_Olambda__174,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_oo(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_174
tff(fact_10303_ATP_Olambda__175,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_op(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_175
tff(fact_10304_ATP_Olambda__176,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_176
tff(fact_10305_ATP_Olambda__177,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_or(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_177
tff(fact_10306_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_zk,Uu),Uua) = cons_nat(Uu,aa_nat_list_nat(nat_list_decode,Uua)) ).

% ATP.lambda_178
tff(fact_10307_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_yy,Uu),Uua))
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_179
tff(fact_10308_ATP_Olambda__180,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_pq,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).

% ATP.lambda_180
tff(fact_10309_ATP_Olambda__181,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_pr(Uu),Uua))
    <=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Uua),Uu)) ) ).

% ATP.lambda_181
tff(fact_10310_ATP_Olambda__182,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_ps,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).

% ATP.lambda_182
tff(fact_10311_ATP_Olambda__183,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_pt,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).

% ATP.lambda_183
tff(fact_10312_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_am,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_184
tff(fact_10313_ATP_Olambda__185,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_pu,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).

% ATP.lambda_185
tff(fact_10314_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_186
tff(fact_10315_ATP_Olambda__187,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_jy,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).

% ATP.lambda_187
tff(fact_10316_ATP_Olambda__188,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ju(Uu),Uua))
    <=> pp(aa_set_nat_bool(ord_less_set_nat(Uua),Uu)) ) ).

% ATP.lambda_188
tff(fact_10317_ATP_Olambda__189,axiom,
    ! [Uu: rat,Uua: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_jv,Uu),Uua))
    <=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).

% ATP.lambda_189
tff(fact_10318_ATP_Olambda__190,axiom,
    ! [Uu: num,Uua: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_jw,Uu),Uua))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).

% ATP.lambda_190
tff(fact_10319_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_al,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_191
tff(fact_10320_ATP_Olambda__192,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_jx,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).

% ATP.lambda_192
tff(fact_10321_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xt,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_193
tff(fact_10322_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_194
tff(fact_10323_ATP_Olambda__195,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xb,Uu),Uua) = aa_nat_real(power_power_real(Uua),Uu) ).

% ATP.lambda_195
tff(fact_10324_ATP_Olambda__196,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aap,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu) ).

% ATP.lambda_196
tff(fact_10325_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_197
tff(fact_10326_ATP_Olambda__198,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_wp(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_198
tff(fact_10327_ATP_Olambda__199,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_xc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_199
tff(fact_10328_ATP_Olambda__200,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_gw,Uu),Uua))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).

% ATP.lambda_200
tff(fact_10329_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_gy,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_201
tff(fact_10330_ATP_Olambda__202,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_gx,Uu),Uua))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).

% ATP.lambda_202
tff(fact_10331_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_wc,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_203
tff(fact_10332_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_204
tff(fact_10333_ATP_Olambda__205,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_xx(Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_205
tff(fact_10334_ATP_Olambda__206,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( pp(aa_complex_bool(aTP_Lamm_a(Uu),Uua))
    <=> pp(aa_set_complex_bool(member_complex(Uua),Uu)) ) ).

% ATP.lambda_206
tff(fact_10335_ATP_Olambda__207,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aa(Uu),Uua))
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_207
tff(fact_10336_ATP_Olambda__208,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( pp(aa_list_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_208
tff(fact_10337_ATP_Olambda__209,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( pp(aa_set_nat_bool(aTP_Lamm_ac(Uu),Uua))
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_209
tff(fact_10338_ATP_Olambda__210,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ad(Uu),Uua))
    <=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).

% ATP.lambda_210
tff(fact_10339_ATP_Olambda__211,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_ae(Uu),Uua))
    <=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).

% ATP.lambda_211
tff(fact_10340_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abe(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_212
tff(fact_10341_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ll(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_213
tff(fact_10342_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lk(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_214
tff(fact_10343_ATP_Olambda__215,axiom,
    ! [Uu: fun_real_bool,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_yr(Uu),Uua))
    <=> pp(aa_real_bool(Uu,aa_real_real(inverse_inverse_real,Uua))) ) ).

% ATP.lambda_215
tff(fact_10344_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cs(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_216
tff(fact_10345_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_yj(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_217
tff(fact_10346_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ia(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_218
tff(fact_10347_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hk(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_219
tff(fact_10348_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hn(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_220
tff(fact_10349_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_aac,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_aab(Uu),Uua)) ).

% ATP.lambda_221
tff(fact_10350_ATP_Olambda__222,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_wk,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wj(Uu),Uua)) ).

% ATP.lambda_222
tff(fact_10351_ATP_Olambda__223,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_wi,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wh(Uu),Uua)) ).

% ATP.lambda_223
tff(fact_10352_ATP_Olambda__224,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_wg,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wf(Uu),Uua)) ).

% ATP.lambda_224
tff(fact_10353_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_we,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wd(Uu),Uua)) ).

% ATP.lambda_225
tff(fact_10354_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_wb,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wa(Uu),Uua)) ).

% ATP.lambda_226
tff(fact_10355_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_xh(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xg(Uu),Uua)) ).

% ATP.lambda_227
tff(fact_10356_ATP_Olambda__228,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_yz(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_228
tff(fact_10357_ATP_Olambda__229,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_yn(Uu),Uua))
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_229
tff(fact_10358_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_iv,Uu),Uua))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ) ).

% ATP.lambda_230
tff(fact_10359_ATP_Olambda__231,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_om(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_231
tff(fact_10360_ATP_Olambda__232,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ol(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_232
tff(fact_10361_ATP_Olambda__233,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ok(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_233
tff(fact_10362_ATP_Olambda__234,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_lt(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_234
tff(fact_10363_ATP_Olambda__235,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_lu(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_235
tff(fact_10364_ATP_Olambda__236,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_lv(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_236
tff(fact_10365_ATP_Olambda__237,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_lx(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_237
tff(fact_10366_ATP_Olambda__238,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_lw(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_238
tff(fact_10367_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_zh(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_239
tff(fact_10368_ATP_Olambda__240,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_yb(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(power_power_real(Uu),Uua)) ).

% ATP.lambda_240
tff(fact_10369_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aav(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_241
tff(fact_10370_ATP_Olambda__242,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aaz(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_242
tff(fact_10371_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_aax(Uu),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)) ).

% ATP.lambda_243
tff(fact_10372_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ww,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_244
tff(fact_10373_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wv,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_245
tff(fact_10374_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vt,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_246
tff(fact_10375_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vs,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_247
tff(fact_10376_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_dc(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_248
tff(fact_10377_ATP_Olambda__249,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_xv(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_249
tff(fact_10378_ATP_Olambda__250,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_zq(Uu),Uua) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_250
tff(fact_10379_ATP_Olambda__251,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_ex(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_251
tff(fact_10380_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ez(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_252
tff(fact_10381_ATP_Olambda__253,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_fd(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_253
tff(fact_10382_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_rat(aTP_Lamm_fa(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_254
tff(fact_10383_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_fc(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_255
tff(fact_10384_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_fb(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_256
tff(fact_10385_ATP_Olambda__257,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_ey(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_257
tff(fact_10386_ATP_Olambda__258,axiom,
    ! [Uu: fun_Pr672294278902479040t_real,Uua: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aTP_Lamm_qn(Uu),Uua) = aa_real_real(ln_ln_real,aa_Pro6168199777849092593t_real(Uu,Uua)) ).

% ATP.lambda_258
tff(fact_10387_ATP_Olambda__259,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: extended_enat] : aa_Ext2541572767299617110t_real(aTP_Lamm_qo(Uu),Uua) = aa_real_real(ln_ln_real,aa_Ext2541572767299617110t_real(Uu,Uua)) ).

% ATP.lambda_259
tff(fact_10388_ATP_Olambda__260,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_qm(Uu),Uua) = aa_real_real(ln_ln_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_260
tff(fact_10389_ATP_Olambda__261,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_qj(Uu),Uua) = aa_real_real(ln_ln_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_261
tff(fact_10390_ATP_Olambda__262,axiom,
    ! [Uu: fun_set_nat_real,Uua: set_nat] : aa_set_nat_real(aTP_Lamm_qk(Uu),Uua) = aa_real_real(ln_ln_real,aa_set_nat_real(Uu,Uua)) ).

% ATP.lambda_262
tff(fact_10391_ATP_Olambda__263,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qp(Uu),Uua) = aa_real_real(ln_ln_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_263
tff(fact_10392_ATP_Olambda__264,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_ql(Uu),Uua) = aa_real_real(ln_ln_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_10393_ATP_Olambda__265,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_zl,Uu),Uua) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_10394_ATP_Olambda__266,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_nd(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_10395_ATP_Olambda__267,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ne(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_10396_ATP_Olambda__268,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_nf(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_10397_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ng(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_269
tff(fact_10398_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_nh(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_10399_ATP_Olambda__271,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_us(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_271
tff(fact_10400_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_complex(aTP_Lamm_um(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_272
tff(fact_10401_ATP_Olambda__273,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_uk(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_273
tff(fact_10402_ATP_Olambda__274,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_ul(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_274
tff(fact_10403_ATP_Olambda__275,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_un(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_275
tff(fact_10404_ATP_Olambda__276,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_complex(aTP_Lamm_uq(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(Uu,Uua)) ).

% ATP.lambda_276
tff(fact_10405_ATP_Olambda__277,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_uo(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_277
tff(fact_10406_ATP_Olambda__278,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_up(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_278
tff(fact_10407_ATP_Olambda__279,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ur(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_279
tff(fact_10408_ATP_Olambda__280,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_pe(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_280
tff(fact_10409_ATP_Olambda__281,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_pb(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_281
tff(fact_10410_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_pd(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_282
tff(fact_10411_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_pc(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_283
tff(fact_10412_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_dd(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_284
tff(fact_10413_ATP_Olambda__285,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fy(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_285
tff(fact_10414_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ye(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_286
tff(fact_10415_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abc(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_al,Uu)) ).

% ATP.lambda_287
tff(fact_10416_ATP_Olambda__288,axiom,
    ! [Uu: set_real,Uua: real] :
      ( pp(aa_real_bool(aTP_Lamm_aad(Uu),Uua))
    <=> ! [X3: real] :
          ( member_real(X3,Uu)
         => pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Uua)) ) ) ).

% ATP.lambda_288
tff(fact_10417_ATP_Olambda__289,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_abf(Uu),Uua))
    <=> ! [X3: nat] :
          ( pp(aa_set_nat_bool(member_nat(X3),Uu))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),X3)) ) ) ).

% ATP.lambda_289
tff(fact_10418_ATP_Olambda__290,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_zr(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,Uu) ).

% ATP.lambda_290
tff(fact_10419_ATP_Olambda__291,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_rat(aTP_Lamm_zu(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,Uu) ).

% ATP.lambda_291
tff(fact_10420_ATP_Olambda__292,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_292
tff(fact_10421_ATP_Olambda__293,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_293
tff(fact_10422_ATP_Olambda__294,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_vb(Uu),Uua),Uub) = if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_294
tff(fact_10423_ATP_Olambda__295,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_vc(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_295
tff(fact_10424_ATP_Olambda__296,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_296
tff(fact_10425_ATP_Olambda__297,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_set_nat_bool(member_nat(Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_297
tff(fact_10426_ATP_Olambda__298,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_cn(Uu),Uua),Uub) = aa_int_int(if_int(aa_set_nat_bool(member_nat(Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_298
tff(fact_10427_ATP_Olambda__299,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_fh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uu),Uub)),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_299
tff(fact_10428_ATP_Olambda__300,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: extended_enat] : aa_Extended_enat_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_fm(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uu),Uub),aa_Extended_enat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_300
tff(fact_10429_ATP_Olambda__301,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_301
tff(fact_10430_ATP_Olambda__302,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fl(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_302
tff(fact_10431_ATP_Olambda__303,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fe(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_303
tff(fact_10432_ATP_Olambda__304,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fi(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_304
tff(fact_10433_ATP_Olambda__305,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_305
tff(fact_10434_ATP_Olambda__306,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fj(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_306
tff(fact_10435_ATP_Olambda__307,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ff(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_307
tff(fact_10436_ATP_Olambda__308,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fk(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_308
tff(fact_10437_ATP_Olambda__309,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex3741802288643809319omplex,Uub: extended_enat] : aa_Ext8748583412541189208omplex(aa_fun616941850244831356omplex(aTP_Lamm_mx(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uu),Uub),aa_Ext8748583412541189208omplex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_309
tff(fact_10438_ATP_Olambda__310,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_nc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uu),Uub)),aa_Ext2541572767299617110t_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_310
tff(fact_10439_ATP_Olambda__311,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_mw(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_311
tff(fact_10440_ATP_Olambda__312,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_nb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_312
tff(fact_10441_ATP_Olambda__313,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_mt(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_313
tff(fact_10442_ATP_Olambda__314,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_my(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_314
tff(fact_10443_ATP_Olambda__315,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mu(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_315
tff(fact_10444_ATP_Olambda__316,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_316
tff(fact_10445_ATP_Olambda__317,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_mv(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_317
tff(fact_10446_ATP_Olambda__318,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_na(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_318
tff(fact_10447_ATP_Olambda__319,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_fr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uub),Uu)),aa_Ext2541572767299617110t_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_319
tff(fact_10448_ATP_Olambda__320,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: extended_enat] : aa_Extended_enat_rat(aa_fun9057213682095532284at_rat(aTP_Lamm_fw(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uub),Uu),aa_Extended_enat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_320
tff(fact_10449_ATP_Olambda__321,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_321
tff(fact_10450_ATP_Olambda__322,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fv(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_322
tff(fact_10451_ATP_Olambda__323,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_323
tff(fact_10452_ATP_Olambda__324,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fs(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_324
tff(fact_10453_ATP_Olambda__325,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_325
tff(fact_10454_ATP_Olambda__326,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_326
tff(fact_10455_ATP_Olambda__327,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ft(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_327
tff(fact_10456_ATP_Olambda__328,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_328
tff(fact_10457_ATP_Olambda__329,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_cd(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_329
tff(fact_10458_ATP_Olambda__330,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_330
tff(fact_10459_ATP_Olambda__331,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fu(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_rat(Uua,Uub)),zero_zero_rat) ).

% ATP.lambda_331
tff(fact_10460_ATP_Olambda__332,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex3741802288643809319omplex,Uub: extended_enat] : aa_Ext8748583412541189208omplex(aa_fun616941850244831356omplex(aTP_Lamm_mn(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uub),Uu),aa_Ext8748583412541189208omplex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_332
tff(fact_10461_ATP_Olambda__333,axiom,
    ! [Uu: extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1749635019461158908t_real(aTP_Lamm_ms(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(Uub),Uu)),aa_Ext2541572767299617110t_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_333
tff(fact_10462_ATP_Olambda__334,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_mm(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_334
tff(fact_10463_ATP_Olambda__335,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_mr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_335
tff(fact_10464_ATP_Olambda__336,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_mj(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_336
tff(fact_10465_ATP_Olambda__337,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_337
tff(fact_10466_ATP_Olambda__338,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mk(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_338
tff(fact_10467_ATP_Olambda__339,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_339
tff(fact_10468_ATP_Olambda__340,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ml(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_complex(Uua,Uub)),one_one_complex) ).

% ATP.lambda_340
tff(fact_10469_ATP_Olambda__341,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_mq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),one_one_real) ).

% ATP.lambda_341
tff(fact_10470_ATP_Olambda__342,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aak(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_342
tff(fact_10471_ATP_Olambda__343,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aai(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_343
tff(fact_10472_ATP_Olambda__344,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aaj(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_344
tff(fact_10473_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ci(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uu,Uub)),aa_nat_real(Uua,Uub)),zero_zero_real) ).

% ATP.lambda_345
tff(fact_10474_ATP_Olambda__346,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(Uu,Uub)),aa_nat_nat(Uua,Uub)),zero_zero_nat) ).

% ATP.lambda_346
tff(fact_10475_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_bool,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ck(Uu),Uua),Uub) = aa_int_int(if_int(aa_nat_bool(Uu,Uub),aa_nat_int(Uua,Uub)),zero_zero_int) ).

% ATP.lambda_347
tff(fact_10476_ATP_Olambda__348,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1090353380788292083t_real(aTP_Lamm_gm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(Uua,Uub)),aa_Ext2541572767299617110t_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_348
tff(fact_10477_ATP_Olambda__349,axiom,
    ! [Uu: fun_Ex4152069839811072081at_rat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Extended_enat_rat(aa_fun6658670135733114143at_rat(aTP_Lamm_gs(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_Ext2772979874139974381t_bool(Uua,Uub),aa_Extended_enat_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_349
tff(fact_10478_ATP_Olambda__350,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_gk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_350
tff(fact_10479_ATP_Olambda__351,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_bool,Uub: complex] : aa_complex_rat(aa_fun5529999419959482889ex_rat(aTP_Lamm_gr(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_complex_bool(Uua,Uub),aa_complex_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_351
tff(fact_10480_ATP_Olambda__352,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_gg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_352
tff(fact_10481_ATP_Olambda__353,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_bool,Uub: real] : aa_real_rat(aa_fun7219699593533116937al_rat(aTP_Lamm_gn(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_real_bool(Uua,Uub),aa_real_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_353
tff(fact_10482_ATP_Olambda__354,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_bool,Uub: real] : aa_real_nat(aa_fun1504556115071737857al_nat(aTP_Lamm_gt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_real_bool(Uua,Uub)),aa_real_nat(Uu,Uub)),zero_zero_nat) ).

% ATP.lambda_354
tff(fact_10483_ATP_Olambda__355,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_bool,Uub: nat] : aa_nat_rat(aa_fun5656162108953708681at_rat(aTP_Lamm_gp(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_nat_bool(Uua,Uub),aa_nat_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_355
tff(fact_10484_ATP_Olambda__356,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_gi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu,Uub)),zero_zero_real) ).

% ATP.lambda_356
tff(fact_10485_ATP_Olambda__357,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_bool,Uub: int] : aa_int_rat(aa_fun4400602662665189257nt_rat(aTP_Lamm_gq(Uu),Uua),Uub) = aa_rat_rat(if_rat(aa_int_bool(Uua,Uub),aa_int_rat(Uu,Uub)),zero_zero_rat) ).

% ATP.lambda_357
tff(fact_10486_ATP_Olambda__358,axiom,
    ! [Uu: fun_Ex3741802288643809319omplex,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Ext8748583412541189208omplex(aa_fun1711458463208245493omplex(aTP_Lamm_nm(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_Ext2772979874139974381t_bool(Uua,Uub),aa_Ext8748583412541189208omplex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_358
tff(fact_10487_ATP_Olambda__359,axiom,
    ! [Uu: fun_Ex1451620480498224165t_real,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] : aa_Ext2541572767299617110t_real(aa_fun1090353380788292083t_real(aTP_Lamm_nr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_Ext2772979874139974381t_bool(Uua,Uub)),aa_Ext2541572767299617110t_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_359
tff(fact_10488_ATP_Olambda__360,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_bool,Uub: complex] : aa_complex_complex(aa_fun1034237174562290143omplex(aTP_Lamm_nl(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(Uua,Uub),aa_complex_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_360
tff(fact_10489_ATP_Olambda__361,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_bool,Uub: complex] : aa_complex_real(aa_fun2902006202505214429x_real(aTP_Lamm_nq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(Uua,Uub)),aa_complex_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_361
tff(fact_10490_ATP_Olambda__362,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_bool,Uub: real] : aa_real_complex(aa_fun6813637976755515871omplex(aTP_Lamm_ni(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(Uua,Uub),aa_real_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_362
tff(fact_10491_ATP_Olambda__363,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_bool,Uub: real] : aa_real_real(aa_fun6348708079608492509l_real(aTP_Lamm_nn(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(Uua,Uub)),aa_real_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_363
tff(fact_10492_ATP_Olambda__364,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_bool,Uub: nat] : aa_nat_complex(aa_fun8588439219086927967omplex(aTP_Lamm_nj(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(Uua,Uub),aa_nat_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_364
tff(fact_10493_ATP_Olambda__365,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_bool,Uub: nat] : aa_nat_real(aa_fun1750223370215434333t_real(aTP_Lamm_no(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(Uua,Uub)),aa_nat_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_365
tff(fact_10494_ATP_Olambda__366,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_bool,Uub: int] : aa_int_complex(aa_fun3606824691168040799omplex(aTP_Lamm_nk(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(Uua,Uub),aa_int_complex(Uu,Uub)),one_one_complex) ).

% ATP.lambda_366
tff(fact_10495_ATP_Olambda__367,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_bool,Uub: int] : aa_int_real(aa_fun6974987029372589917t_real(aTP_Lamm_np(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(Uua,Uub)),aa_int_real(Uu,Uub)),one_one_real) ).

% ATP.lambda_367
tff(fact_10496_ATP_Olambda__368,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_yo(Uu),Uua),Uub))
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_368
tff(fact_10497_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qa(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_369
tff(fact_10498_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_370
tff(fact_10499_ATP_Olambda__371,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qf(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_371
tff(fact_10500_ATP_Olambda__372,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uy(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ux(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_372
tff(fact_10501_ATP_Olambda__373,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uw(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_uv(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_373
tff(fact_10502_ATP_Olambda__374,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uj(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ui(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_374
tff(fact_10503_ATP_Olambda__375,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uh(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ug(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_375
tff(fact_10504_ATP_Olambda__376,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uf(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ue(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_376
tff(fact_10505_ATP_Olambda__377,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ud(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_uc(Uu),Uua),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_377
tff(fact_10506_ATP_Olambda__378,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hx(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_complex),zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_378
tff(fact_10507_ATP_Olambda__379,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_real),zero_zero_real)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_379
tff(fact_10508_ATP_Olambda__380,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hz(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_int),zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_380
tff(fact_10509_ATP_Olambda__381,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_aah(Uu),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).

% ATP.lambda_381
tff(fact_10510_ATP_Olambda__382,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qg(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_qf(Uu),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_382
tff(fact_10511_ATP_Olambda__383,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qd(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_383
tff(fact_10512_ATP_Olambda__384,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qb(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qa(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_384
tff(fact_10513_ATP_Olambda__385,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_py(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_px(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_385
tff(fact_10514_ATP_Olambda__386,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_vn(Uu),Uua),Uub))
    <=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_vm(Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_386
tff(fact_10515_ATP_Olambda__387,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_387
tff(fact_10516_ATP_Olambda__388,axiom,
    ! [Uu: real,Uua: fun_na2180513548729900188l_real,Uub: nat] : aa_nat_real(aa_fun4674562053777771865t_real(aTP_Lamm_kc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_literal_real(aa_nat8303615512603269253l_real(Uua,Uub),zero_zero_literal)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_388
tff(fact_10517_ATP_Olambda__389,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_kd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_389
tff(fact_10518_ATP_Olambda__390,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_ke(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_390
tff(fact_10519_ATP_Olambda__391,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_kf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_391
tff(fact_10520_ATP_Olambda__392,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_kg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_392
tff(fact_10521_ATP_Olambda__393,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_qv(Uu),Uua),Uub))
    <=> ( groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_cv(Uu),Uub),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_393
tff(fact_10522_ATP_Olambda__394,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( pp(aa_real_bool(aa_nat_fun_real_bool(aTP_Lamm_qw(Uu),Uua),Uub))
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_394
tff(fact_10523_ATP_Olambda__395,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gb(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_10524_ATP_Olambda__396,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_396
tff(fact_10525_ATP_Olambda__397,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ga(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_397
tff(fact_10526_ATP_Olambda__398,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_398
tff(fact_10527_ATP_Olambda__399,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_kh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_399
tff(fact_10528_ATP_Olambda__400,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ze(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_400
tff(fact_10529_ATP_Olambda__401,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jo(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_401
tff(fact_10530_ATP_Olambda__402,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_402
tff(fact_10531_ATP_Olambda__403,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_zf(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_403
tff(fact_10532_ATP_Olambda__404,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_bl(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua)) ) ) ).

% ATP.lambda_404
tff(fact_10533_ATP_Olambda__405,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_bj(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s5460976970255530739at_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_405
tff(fact_10534_ATP_Olambda__406,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( pp(aa_lis1364384421681148007t_bool(aa_nat4017767907309459068t_bool(aTP_Lamm_bk(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3941691890525107288d_enat(Uub)),Uua)) ) ) ).

% ATP.lambda_406
tff(fact_10535_ATP_Olambda__407,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_bi(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).

% ATP.lambda_407
tff(fact_10536_ATP_Olambda__408,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bm(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Uub)),Uua)) ) ) ).

% ATP.lambda_408
tff(fact_10537_ATP_Olambda__409,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_bn(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).

% ATP.lambda_409
tff(fact_10538_ATP_Olambda__410,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_bh(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Uub)),Uua)) ) ) ).

% ATP.lambda_410
tff(fact_10539_ATP_Olambda__411,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_be(Uu),Uua),Uub))
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_411
tff(fact_10540_ATP_Olambda__412,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: list_P6011104703257516679at_nat] :
      ( pp(aa_lis6649954313659033868t_bool(aa_nat1772366067986465313t_bool(aTP_Lamm_bc(Uu),Uua),Uub))
    <=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Uub)),Uu))
        & ( size_s5460976970255530739at_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_412
tff(fact_10541_ATP_Olambda__413,axiom,
    ! [Uu: set_Extended_enat,Uua: nat,Uub: list_Extended_enat] :
      ( pp(aa_lis1364384421681148007t_bool(aa_nat4017767907309459068t_bool(aTP_Lamm_bd(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(ord_le7203529160286727270d_enat(set_Extended_enat2(Uub)),Uu))
        & ( size_s3941691890525107288d_enat(Uub) = Uua ) ) ) ).

% ATP.lambda_413
tff(fact_10542_ATP_Olambda__414,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_bb(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_414
tff(fact_10543_ATP_Olambda__415,axiom,
    ! [Uu: set_bool,Uua: nat,Uub: list_bool] :
      ( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_bf(Uu),Uua),Uub))
    <=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
        & ( size_size_list_bool(Uub) = Uua ) ) ) ).

% ATP.lambda_415
tff(fact_10544_ATP_Olambda__416,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_bg(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_416
tff(fact_10545_ATP_Olambda__417,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_ba(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_417
tff(fact_10546_ATP_Olambda__418,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_zd(Uu),Uua),Uub))
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_418
tff(fact_10547_ATP_Olambda__419,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_vw(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_419
tff(fact_10548_ATP_Olambda__420,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( pp(aa_set_nat_bool(aa_nat8857836526276733660t_bool(aTP_Lamm_aaq(Uu),Uua),Uub))
    <=> ( member_set_nat(Uub,pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_420
tff(fact_10549_ATP_Olambda__421,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_abd(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_421
tff(fact_10550_ATP_Olambda__422,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_vx(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_422
tff(fact_10551_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jn(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_423
tff(fact_10552_ATP_Olambda__424,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_424
tff(fact_10553_ATP_Olambda__425,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_li(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_425
tff(fact_10554_ATP_Olambda__426,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_426
tff(fact_10555_ATP_Olambda__427,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_427
tff(fact_10556_ATP_Olambda__428,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_vg(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_428
tff(fact_10557_ATP_Olambda__429,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_af(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_429
tff(fact_10558_ATP_Olambda__430,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_vh(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).

% ATP.lambda_430
tff(fact_10559_ATP_Olambda__431,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ak(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_431
tff(fact_10560_ATP_Olambda__432,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_aj(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).

% ATP.lambda_432
tff(fact_10561_ATP_Olambda__433,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ag(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).

% ATP.lambda_433
tff(fact_10562_ATP_Olambda__434,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ah(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).

% ATP.lambda_434
tff(fact_10563_ATP_Olambda__435,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ai(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_435
tff(fact_10564_ATP_Olambda__436,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_wt(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).

% ATP.lambda_436
tff(fact_10565_ATP_Olambda__437,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_wr(Uu),Uua),Uub))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uua))
        & pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uub),Uu)) ) ) ).

% ATP.lambda_437
tff(fact_10566_ATP_Olambda__438,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex8563670519792538868t_bool,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun4759809662886658982t_bool(aTP_Lamm_gl(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & pp(aa_Ext2772979874139974381t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_438
tff(fact_10567_ATP_Olambda__439,axiom,
    ! [Uu: set_complex,Uua: fun_complex_bool,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun1096865461735438652x_bool(aTP_Lamm_gj(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & pp(aa_complex_bool(Uua,Uub)) ) ) ).

% ATP.lambda_439
tff(fact_10568_ATP_Olambda__440,axiom,
    ! [Uu: set_real,Uua: fun_real_bool,Uub: real] :
      ( pp(aa_real_bool(aa_fun6476784834714468668l_bool(aTP_Lamm_gf(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & pp(aa_real_bool(Uua,Uub)) ) ) ).

% ATP.lambda_440
tff(fact_10569_ATP_Olambda__441,axiom,
    ! [Uu: set_nat,Uua: fun_nat_bool,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun3878383735219268284t_bool(aTP_Lamm_go(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & pp(aa_nat_bool(Uua,Uub)) ) ) ).

% ATP.lambda_441
tff(fact_10570_ATP_Olambda__442,axiom,
    ! [Uu: set_int,Uua: fun_int_bool,Uub: int] :
      ( pp(aa_int_bool(aa_fun5146555767781538748t_bool(aTP_Lamm_gh(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & pp(aa_int_bool(Uua,Uub)) ) ) ).

% ATP.lambda_442
tff(fact_10571_ATP_Olambda__443,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aTP_Lamm_ei(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Ext2541572767299617110t_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_443
tff(fact_10572_ATP_Olambda__444,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun5314330785533781769t_bool(aTP_Lamm_es(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Extended_enat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_444
tff(fact_10573_ATP_Olambda__445,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_eg(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_445
tff(fact_10574_ATP_Olambda__446,axiom,
    ! [Uu: set_complex,Uua: fun_complex_rat,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2759627087638585025x_bool(aTP_Lamm_eq(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_446
tff(fact_10575_ATP_Olambda__447,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_ea(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_447
tff(fact_10576_ATP_Olambda__448,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_ek(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_448
tff(fact_10577_ATP_Olambda__449,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ec(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_449
tff(fact_10578_ATP_Olambda__450,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_em(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_450
tff(fact_10579_ATP_Olambda__451,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_ee(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_451
tff(fact_10580_ATP_Olambda__452,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aTP_Lamm_eo(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_452
tff(fact_10581_ATP_Olambda__453,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3741802288643809319omplex,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun7434105802671982559t_bool(aTP_Lamm_do(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Ext8748583412541189208omplex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_453
tff(fact_10582_ATP_Olambda__454,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aTP_Lamm_dy(Uu),Uua),Uub))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uub),Uu))
        & ( aa_Ext2541572767299617110t_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_454
tff(fact_10583_ATP_Olambda__455,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_dm(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_455
tff(fact_10584_ATP_Olambda__456,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_dw(Uu),Uua),Uub))
    <=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_456
tff(fact_10585_ATP_Olambda__457,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_dg(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_457
tff(fact_10586_ATP_Olambda__458,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_dq(Uu),Uua),Uub))
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_458
tff(fact_10587_ATP_Olambda__459,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_di(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_459
tff(fact_10588_ATP_Olambda__460,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_ds(Uu),Uua),Uub))
    <=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_460
tff(fact_10589_ATP_Olambda__461,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_dk(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_461
tff(fact_10590_ATP_Olambda__462,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_du(Uu),Uua),Uub))
    <=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_462
tff(fact_10591_ATP_Olambda__463,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ws(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).

% ATP.lambda_463
tff(fact_10592_ATP_Olambda__464,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_zg(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).

% ATP.lambda_464
tff(fact_10593_ATP_Olambda__465,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_aan(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu)) ) ).

% ATP.lambda_465
tff(fact_10594_ATP_Olambda__466,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_aal(Uu),Uua),Uub))
    <=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu)) ) ).

% ATP.lambda_466
tff(fact_10595_ATP_Olambda__467,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_bo(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_467
tff(fact_10596_ATP_Olambda__468,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_vy(Uu),Uua),Uub))
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_468
tff(fact_10597_ATP_Olambda__469,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ay(Uu),Uua),Uub))
    <=> ( member_real(Uub,ring_1_Ints_real)
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),Uub))
        & pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uub),Uua)) ) ) ).

% ATP.lambda_469
tff(fact_10598_ATP_Olambda__470,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] :
      ( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_az(Uu),Uua),Uub))
    <=> ( member_rat(Uub,ring_1_Ints_rat)
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uu),Uub))
        & pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uub),Uua)) ) ) ).

% ATP.lambda_470
tff(fact_10599_ATP_Olambda__471,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_cy(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_471
tff(fact_10600_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_472
tff(fact_10601_ATP_Olambda__473,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_yf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_473
tff(fact_10602_ATP_Olambda__474,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_474
tff(fact_10603_ATP_Olambda__475,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_cv(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_475
tff(fact_10604_ATP_Olambda__476,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ct(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub)) ).

% ATP.lambda_476
tff(fact_10605_ATP_Olambda__477,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_qx(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_477
tff(fact_10606_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_478
tff(fact_10607_ATP_Olambda__479,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_qy(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(power_power_int(Uua),Uub)) ).

% ATP.lambda_479
tff(fact_10608_ATP_Olambda__480,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_aw(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_480
tff(fact_10609_ATP_Olambda__481,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_481
tff(fact_10610_ATP_Olambda__482,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_482
tff(fact_10611_ATP_Olambda__483,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zp,Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_483
tff(fact_10612_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_484
tff(fact_10613_ATP_Olambda__485,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_ie(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_485
tff(fact_10614_ATP_Olambda__486,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ii(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_486
tff(fact_10615_ATP_Olambda__487,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_il(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_487
tff(fact_10616_ATP_Olambda__488,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_if(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_488
tff(fact_10617_ATP_Olambda__489,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ij(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_489
tff(fact_10618_ATP_Olambda__490,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_im(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_490
tff(fact_10619_ATP_Olambda__491,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_ig(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_491
tff(fact_10620_ATP_Olambda__492,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_in(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_492
tff(fact_10621_ATP_Olambda__493,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_ih(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_493
tff(fact_10622_ATP_Olambda__494,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ik(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_494
tff(fact_10623_ATP_Olambda__495,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_495
tff(fact_10624_ATP_Olambda__496,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_zo(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_496
tff(fact_10625_ATP_Olambda__497,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_jz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).

% ATP.lambda_497
tff(fact_10626_ATP_Olambda__498,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_js(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).

% ATP.lambda_498
tff(fact_10627_ATP_Olambda__499,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ka(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu,Uub)) ).

% ATP.lambda_499
tff(fact_10628_ATP_Olambda__500,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_500
tff(fact_10629_ATP_Olambda__501,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_jt(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_501
tff(fact_10630_ATP_Olambda__502,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_kb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).

% ATP.lambda_502
tff(fact_10631_ATP_Olambda__503,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_gd(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_503
tff(fact_10632_ATP_Olambda__504,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_504
tff(fact_10633_ATP_Olambda__505,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_zn,Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_505
tff(fact_10634_ATP_Olambda__506,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_cq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_506
tff(fact_10635_ATP_Olambda__507,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_cr(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_507
tff(fact_10636_ATP_Olambda__508,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ge(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_508
tff(fact_10637_ATP_Olambda__509,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_xe(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_509
tff(fact_10638_ATP_Olambda__510,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ax(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).

% ATP.lambda_510
tff(fact_10639_ATP_Olambda__511,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ch(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_511
tff(fact_10640_ATP_Olambda__512,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_512
tff(fact_10641_ATP_Olambda__513,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_513
tff(fact_10642_ATP_Olambda__514,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_xa(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_514
tff(fact_10643_ATP_Olambda__515,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yi(Uu),Uua),Uub) = aa_nat_real(power_power_real(aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_515
tff(fact_10644_ATP_Olambda__516,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_xd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_516
tff(fact_10645_ATP_Olambda__517,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_df(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_517
tff(fact_10646_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_zv(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu,Uub))),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uua,Uub))) ).

% ATP.lambda_518
tff(fact_10647_ATP_Olambda__519,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_ja(Uu),Uua),Uub) = aa_Ext7969882492699152700d_enat(plus_p3455044024723400733d_enat(Uu),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub)),Uua)) ).

% ATP.lambda_519
tff(fact_10648_ATP_Olambda__520,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jb(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_520
tff(fact_10649_ATP_Olambda__521,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_521
tff(fact_10650_ATP_Olambda__522,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_jc(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_522
tff(fact_10651_ATP_Olambda__523,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_je(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).

% ATP.lambda_523
tff(fact_10652_ATP_Olambda__524,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_jd(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).

% ATP.lambda_524
tff(fact_10653_ATP_Olambda__525,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ld(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_525
tff(fact_10654_ATP_Olambda__526,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lg(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_526
tff(fact_10655_ATP_Olambda__527,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_le(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_527
tff(fact_10656_ATP_Olambda__528,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lf(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_528
tff(fact_10657_ATP_Olambda__529,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ya(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_529
tff(fact_10658_ATP_Olambda__530,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_hp(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_530
tff(fact_10659_ATP_Olambda__531,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hs(Uu),Uua),Uub) = aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_531
tff(fact_10660_ATP_Olambda__532,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_hq(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_532
tff(fact_10661_ATP_Olambda__533,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_hr(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_533
tff(fact_10662_ATP_Olambda__534,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_534
tff(fact_10663_ATP_Olambda__535,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_yl(Uu),Uua),Uub))
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_535
tff(fact_10664_ATP_Olambda__536,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hm(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_536
tff(fact_10665_ATP_Olambda__537,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_537
tff(fact_10666_ATP_Olambda__538,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_os(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_538
tff(fact_10667_ATP_Olambda__539,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ot(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_539
tff(fact_10668_ATP_Olambda__540,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ou(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_540
tff(fact_10669_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_541
tff(fact_10670_ATP_Olambda__542,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ow(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_542
tff(fact_10671_ATP_Olambda__543,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_aat(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real2,Uu),Uua) ).

% ATP.lambda_543
tff(fact_10672_ATP_Olambda__544,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_kj(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_544
tff(fact_10673_ATP_Olambda__545,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ki(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_545
tff(fact_10674_ATP_Olambda__546,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_od(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_546
tff(fact_10675_ATP_Olambda__547,axiom,
    ! [Uu: fun_real_bool,Uua: real,Uub: real] :
      ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ym(Uu),Uua),Uub))
    <=> pp(aa_real_bool(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua))) ) ).

% ATP.lambda_547
tff(fact_10676_ATP_Olambda__548,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_de(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_548
tff(fact_10677_ATP_Olambda__549,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ce(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_549
tff(fact_10678_ATP_Olambda__550,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_yk(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_550
tff(fact_10679_ATP_Olambda__551,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hl(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_551
tff(fact_10680_ATP_Olambda__552,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ns(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_552
tff(fact_10681_ATP_Olambda__553,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ho(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_553
tff(fact_10682_ATP_Olambda__554,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_za(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_554
tff(fact_10683_ATP_Olambda__555,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_cw(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).

% ATP.lambda_555
tff(fact_10684_ATP_Olambda__556,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cu(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(power_power_real(Uua),Uub))) ).

% ATP.lambda_556
tff(fact_10685_ATP_Olambda__557,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_uv(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc)),aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_557
tff(fact_10686_ATP_Olambda__558,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ux(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_real) ).

% ATP.lambda_558
tff(fact_10687_ATP_Olambda__559,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_uc(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc))),aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_559
tff(fact_10688_ATP_Olambda__560,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ue(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_real) ).

% ATP.lambda_560
tff(fact_10689_ATP_Olambda__561,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ug(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_complex) ).

% ATP.lambda_561
tff(fact_10690_ATP_Olambda__562,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ui(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),zero_zero_real) ).

% ATP.lambda_562
tff(fact_10691_ATP_Olambda__563,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),zero_zero_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_563
tff(fact_10692_ATP_Olambda__564,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rw(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_zero_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_564
tff(fact_10693_ATP_Olambda__565,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sa(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),zero_zero_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_565
tff(fact_10694_ATP_Olambda__566,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),zero_zero_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_566
tff(fact_10695_ATP_Olambda__567,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_se(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_complex),aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_567
tff(fact_10696_ATP_Olambda__568,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_real),aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_568
tff(fact_10697_ATP_Olambda__569,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_sh(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_rat),aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_569
tff(fact_10698_ATP_Olambda__570,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_si(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_nat),aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_570
tff(fact_10699_ATP_Olambda__571,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_sj(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_int),aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat))))) ).

% ATP.lambda_571
tff(fact_10700_ATP_Olambda__572,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_sf(Uu),Uua),Uub),Uuc) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_572
tff(fact_10701_ATP_Olambda__573,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_sd(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_573
tff(fact_10702_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_574
tff(fact_10703_ATP_Olambda__575,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_sb(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu)),aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_575
tff(fact_10704_ATP_Olambda__576,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc)),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_576
tff(fact_10705_ATP_Olambda__577,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_gv(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uuc),Uua)),aa_nat_real(Uub,Uuc)),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_577
tff(fact_10706_ATP_Olambda__578,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jm(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jl(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_578
tff(fact_10707_ATP_Olambda__579,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jk(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jj(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_579
tff(fact_10708_ATP_Olambda__580,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jh(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).

% ATP.lambda_580
tff(fact_10709_ATP_Olambda__581,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_st(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),if_Code_integer(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_Code_integer,zero_z3403309356797280102nteger))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_581
tff(fact_10710_ATP_Olambda__582,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_sr(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_complex_complex(uminus1482373934393186551omplex,Uub)),aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_complex),zero_zero_complex))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_582
tff(fact_10711_ATP_Olambda__583,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_su(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat)),aa_real_real(uminus_uminus_real,Uub)),aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu)),one_one_real),zero_zero_real))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_583
tff(fact_10712_ATP_Olambda__584,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ss(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_rat_rat(uminus_uminus_rat,Uub)),aa_rat_rat(if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_rat),zero_zero_rat))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_584
tff(fact_10713_ATP_Olambda__585,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_sq(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),zero_zero_nat),aa_int_int(uminus_uminus_int,Uub)),aa_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uuc),Uu),one_one_int),zero_zero_int))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_585
tff(fact_10714_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_rv(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ru(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_586
tff(fact_10715_ATP_Olambda__587,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_rh(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_rg(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_587
tff(fact_10716_ATP_Olambda__588,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_rf(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_re(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_588
tff(fact_10717_ATP_Olambda__589,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_rd(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_rc(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_589
tff(fact_10718_ATP_Olambda__590,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_rb(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_ra(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_590
tff(fact_10719_ATP_Olambda__591,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xn(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_591
tff(fact_10720_ATP_Olambda__592,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xl(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_592
tff(fact_10721_ATP_Olambda__593,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xj(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_593
tff(fact_10722_ATP_Olambda__594,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_xk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_594
tff(fact_10723_ATP_Olambda__595,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kn(Uu),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uu),Uub)) ).

% ATP.lambda_595
tff(fact_10724_ATP_Olambda__596,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_kt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uu),Uub)) ).

% ATP.lambda_596
tff(fact_10725_ATP_Olambda__597,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kp(Uu),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uu),Uub)) ).

% ATP.lambda_597
tff(fact_10726_ATP_Olambda__598,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_kr(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uu),Uub)) ).

% ATP.lambda_598
tff(fact_10727_ATP_Olambda__599,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wa(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_599
tff(fact_10728_ATP_Olambda__600,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jh(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_600
tff(fact_10729_ATP_Olambda__601,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jl(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_601
tff(fact_10730_ATP_Olambda__602,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_jj(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uua),Uu)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_602
tff(fact_10731_ATP_Olambda__603,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rp(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uu),Uuc))),aa_nat_complex(comm_s2602460028002588243omplex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_603
tff(fact_10732_ATP_Olambda__604,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rs(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uu),Uuc))),aa_nat_real(comm_s7457072308508201937r_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_604
tff(fact_10733_ATP_Olambda__605,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rq(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_605
tff(fact_10734_ATP_Olambda__606,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rr(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uu),Uuc))),aa_nat_int(comm_s4660882817536571857er_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_606
tff(fact_10735_ATP_Olambda__607,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rl(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_607
tff(fact_10736_ATP_Olambda__608,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ro(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(power_power_real(Uu),Uuc))),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_608
tff(fact_10737_ATP_Olambda__609,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rm(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_609
tff(fact_10738_ATP_Olambda__610,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qz(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_610
tff(fact_10739_ATP_Olambda__611,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rn(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_611
tff(fact_10740_ATP_Olambda__612,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_kv(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu),Uuc)) ).

% ATP.lambda_612
tff(fact_10741_ATP_Olambda__613,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ky(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(power_power_real(Uu),Uuc)) ).

% ATP.lambda_613
tff(fact_10742_ATP_Olambda__614,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_kw(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).

% ATP.lambda_614
tff(fact_10743_ATP_Olambda__615,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_kx(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu),Uuc)) ).

% ATP.lambda_615
tff(fact_10744_ATP_Olambda__616,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ko(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_616
tff(fact_10745_ATP_Olambda__617,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ku(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(power_power_real(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_617
tff(fact_10746_ATP_Olambda__618,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_kq(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_618
tff(fact_10747_ATP_Olambda__619,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ks(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_619
tff(fact_10748_ATP_Olambda__620,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_kz(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_620
tff(fact_10749_ATP_Olambda__621,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(power_power_real(Uu),Uuc)),aa_nat_real(power_power_real(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_621
tff(fact_10750_ATP_Olambda__622,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_la(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_622
tff(fact_10751_ATP_Olambda__623,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_lb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_623
tff(fact_10752_ATP_Olambda__624,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_qu(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_624
tff(fact_10753_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wf(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_625
tff(fact_10754_ATP_Olambda__626,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_vm(Uu),Uua),Uub),Uuc))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).

% ATP.lambda_626
tff(fact_10755_ATP_Olambda__627,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_wd(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_627
tff(fact_10756_ATP_Olambda__628,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wh(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_628
tff(fact_10757_ATP_Olambda__629,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_wj(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_629
tff(fact_10758_ATP_Olambda__630,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_aab(Uu),Uua),Uub),Uuc))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_630
tff(fact_10759_ATP_Olambda__631,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex3741802288643809319omplex,Uub: fun_Ex3741802288643809319omplex,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun7434105802671982559t_bool(aa_fun6445447259835133315t_bool(aTP_Lamm_dp(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_Ext8748583412541189208omplex(Uua,Uuc)),aa_Ext8748583412541189208omplex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_631
tff(fact_10760_ATP_Olambda__632,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_dz(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_632
tff(fact_10761_ATP_Olambda__633,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_dn(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_633
tff(fact_10762_ATP_Olambda__634,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_dx(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_634
tff(fact_10763_ATP_Olambda__635,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_dh(Uu),Uua),Uub),Uuc))
    <=> ( member_real(Uuc,Uu)
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_635
tff(fact_10764_ATP_Olambda__636,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_dr(Uu),Uua),Uub),Uuc))
    <=> ( member_real(Uuc,Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_636
tff(fact_10765_ATP_Olambda__637,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_dj(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_637
tff(fact_10766_ATP_Olambda__638,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_dt(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_638
tff(fact_10767_ATP_Olambda__639,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_dl(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_639
tff(fact_10768_ATP_Olambda__640,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_dv(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_640
tff(fact_10769_ATP_Olambda__641,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex1451620480498224165t_real,Uub: fun_Ex1451620480498224165t_real,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun338017137003533149t_bool(aa_fun6293361260138588547t_bool(aTP_Lamm_ej(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Ext2541572767299617110t_real(Uua,Uuc)),aa_Ext2541572767299617110t_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_641
tff(fact_10770_ATP_Olambda__642,axiom,
    ! [Uu: set_Extended_enat,Uua: fun_Ex4152069839811072081at_rat,Uub: fun_Ex4152069839811072081at_rat,Uuc: extended_enat] :
      ( pp(aa_Ext2772979874139974381t_bool(aa_fun5314330785533781769t_bool(aa_fun4769541129632704515t_bool(aTP_Lamm_et(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set3347977610044113549t_bool(member_Extended_enat(Uuc),Uu))
        & ( aa_rat_rat(plus_plus_rat(aa_Extended_enat_rat(Uua,Uuc)),aa_Extended_enat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_642
tff(fact_10771_ATP_Olambda__643,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_eh(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_643
tff(fact_10772_ATP_Olambda__644,axiom,
    ! [Uu: set_complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      ( pp(aa_complex_bool(aa_fun2759627087638585025x_bool(aa_fun1496239077807485399x_bool(aTP_Lamm_er(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
        & ( aa_rat_rat(plus_plus_rat(aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_644
tff(fact_10773_ATP_Olambda__645,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_eb(Uu),Uua),Uub),Uuc))
    <=> ( member_real(Uuc,Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_645
tff(fact_10774_ATP_Olambda__646,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_el(Uu),Uua),Uub),Uuc))
    <=> ( member_real(Uuc,Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_646
tff(fact_10775_ATP_Olambda__647,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_ed(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_647
tff(fact_10776_ATP_Olambda__648,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_en(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_648
tff(fact_10777_ATP_Olambda__649,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ef(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_649
tff(fact_10778_ATP_Olambda__650,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( pp(aa_int_bool(aa_fun258058769140594373t_bool(aa_fun465274253410646749t_bool(aTP_Lamm_ep(Uu),Uua),Uub),Uuc))
    <=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
        & ( aa_rat_rat(plus_plus_rat(aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_650
tff(fact_10779_ATP_Olambda__651,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_da(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_651
tff(fact_10780_ATP_Olambda__652,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_db(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(power_power_real(Uub),Uuc)) ).

% ATP.lambda_652
tff(fact_10781_ATP_Olambda__653,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tu(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_653
tff(fact_10782_ATP_Olambda__654,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ua(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(power_power_real(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_654
tff(fact_10783_ATP_Olambda__655,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tw(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_655
tff(fact_10784_ATP_Olambda__656,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ty(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_656
tff(fact_10785_ATP_Olambda__657,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_ra(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_657
tff(fact_10786_ATP_Olambda__658,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_rg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_658
tff(fact_10787_ATP_Olambda__659,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_rc(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_659
tff(fact_10788_ATP_Olambda__660,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ru(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_660
tff(fact_10789_ATP_Olambda__661,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_re(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_661
tff(fact_10790_ATP_Olambda__662,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_hv(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_662
tff(fact_10791_ATP_Olambda__663,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_hu(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_663
tff(fact_10792_ATP_Olambda__664,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_664
tff(fact_10793_ATP_Olambda__665,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_tn(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_tm(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_665
tff(fact_10794_ATP_Olambda__666,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_tl(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_tk(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_666
tff(fact_10795_ATP_Olambda__667,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_tj(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_ti(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_667
tff(fact_10796_ATP_Olambda__668,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_th(Uu),Uua),Uub),Uuc),Uud) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_tg(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_668
tff(fact_10797_ATP_Olambda__669,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_xm(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_xl(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(power_power_real(Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_669
tff(fact_10798_ATP_Olambda__670,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_ub(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ua(Uua),Uuc),Uud)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_real(power_power_real(Uub),Uud)) ).

% ATP.lambda_670
tff(fact_10799_ATP_Olambda__671,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_tz(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ty(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_671
tff(fact_10800_ATP_Olambda__672,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_tx(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_tw(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_672
tff(fact_10801_ATP_Olambda__673,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_tv(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_tu(Uua),Uuc),Uud),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud)),Uu))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_673
tff(fact_10802_ATP_Olambda__674,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_td(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_674
tff(fact_10803_ATP_Olambda__675,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_tf(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_675
tff(fact_10804_ATP_Olambda__676,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_te(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_676
tff(fact_10805_ATP_Olambda__677,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_sk(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_677
tff(fact_10806_ATP_Olambda__678,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_so(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_678
tff(fact_10807_ATP_Olambda__679,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_sm(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_679
tff(fact_10808_ATP_Olambda__680,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_sl(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_680
tff(fact_10809_ATP_Olambda__681,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_sp(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(gbinomial_real(aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(power_power_real(aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(power_power_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_681
tff(fact_10810_ATP_Olambda__682,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_sn(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,Uua)),Uud)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_682
tff(fact_10811_ATP_Olambda__683,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_tg(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_683
tff(fact_10812_ATP_Olambda__684,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: real,Uuc: nat,Uud: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_tm(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(power_power_real(Uub),Uud))),aa_nat_real(power_power_real(Uua),Uuc)) ).

% ATP.lambda_684
tff(fact_10813_ATP_Olambda__685,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: rat,Uuc: nat,Uud: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_ti(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_685
tff(fact_10814_ATP_Olambda__686,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: int,Uuc: nat,Uud: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_tk(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_686
tff(fact_10815_ATP_Olambda__687,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_yu(Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
        & ! [I3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))
           => ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I3)),X_13))
            <=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I3)) ) )
        & ( ( Uuc = Uud )
         => ! [X3: vEBT_VEBT] :
              ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Uua)))
             => ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),Uud))
            & ! [X3: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
               => ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),X3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X3))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_687
tff(fact_10816_ATP_Olambda__688,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aTP_Lamm_co(Uu),Uua) = Uu ).

% ATP.lambda_688
tff(fact_10817_ATP_Olambda__689,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_zm(Uu),Uua) = Uu ).

% ATP.lambda_689
tff(fact_10818_ATP_Olambda__690,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ar,Uu) = Uu ).

% ATP.lambda_690
tff(fact_10819_ATP_Olambda__691,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_as,Uu) = Uu ).

% ATP.lambda_691
tff(fact_10820_ATP_Olambda__692,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_at,Uu) = Uu ).

% ATP.lambda_692
tff(fact_10821_ATP_Olambda__693,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_au,Uu) = Uu ).

% ATP.lambda_693
tff(fact_10822_ATP_Olambda__694,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_av,Uu) = Uu ).

% ATP.lambda_694
tff(fact_10823_ATP_Olambda__695,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ew,Uu) = zero_zero_complex ).

% ATP.lambda_695
tff(fact_10824_ATP_Olambda__696,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_an,Uu) = zero_zero_real ).

% ATP.lambda_696
tff(fact_10825_ATP_Olambda__697,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ao,Uu) = zero_zero_rat ).

% ATP.lambda_697
tff(fact_10826_ATP_Olambda__698,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_bw,Uu) = zero_zero_real ).

% ATP.lambda_698
tff(fact_10827_ATP_Olambda__699,axiom,
    ! [Uu: nat] : aa_nat_rat(aTP_Lamm_zs,Uu) = zero_zero_rat ).

% ATP.lambda_699
tff(fact_10828_ATP_Olambda__700,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ap,Uu) = zero_zero_nat ).

% ATP.lambda_700
tff(fact_10829_ATP_Olambda__701,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_bx,Uu) = zero_zero_int ).

% ATP.lambda_701
tff(fact_10830_ATP_Olambda__702,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aq,Uu) = zero_zero_int ).

% ATP.lambda_702
tff(fact_10831_ATP_Olambda__703,axiom,
    ! [Uu: nat] : aa_nat_rat(aTP_Lamm_zt,Uu) = one_one_rat ).

% ATP.lambda_703
tff(fact_10832_ATP_Olambda__704,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_mg,Uu) = one_one_nat ).

% ATP.lambda_704
tff(fact_10833_ATP_Olambda__705,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_mh,Uu) = one_one_int ).

% ATP.lambda_705
tff(fact_10834_ATP_Olambda__706,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_mi,Uu) = one_one_int ).

% ATP.lambda_706
tff(fact_10835_ATP_Olambda__707,axiom,
    ! [Uu: real] :
      ( pp(aa_real_bool(aTP_Lamm_vj,Uu))
    <=> $false ) ).

% ATP.lambda_707
tff(fact_10836_ATP_Olambda__708,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_vr,Uu))
    <=> $false ) ).

% ATP.lambda_708
tff(fact_10837_ATP_Olambda__709,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_vq,Uu))
    <=> $true ) ).

% ATP.lambda_709

% Helper facts (66)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fconj(P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(fconj(P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(fdisj(P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(fdisj(P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(if_int(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(if_num(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] : aa_num_num(if_num(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(if_rat(fTrue,X),Y) = X ).

tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
    ! [P: fun_nat_bool,X: nat] :
      ( ~ pp(aa_nat_bool(P,X))
      | pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).

tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      | pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_bool] : aa_real_bool(P,fChoice_real(P)) = aa_fun3352386287069783311l_bool(fEx_real,P) ).

tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( X != Y )
      | pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(if_Extended_enat(fTrue,X),Y) = X ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : if_Code_integer(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      | pp(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ pp(aa_Ext2772979874139974381t_bool(fequal_Extended_enat(X),Y))
      | ( X = Y ) ) ).

tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
      ( ~ pp(fAll_VEBT_VEBT(P))
      | pp(aa_VEBT_VEBT_bool(P,X)) ) ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] : if_option_nat(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
    ! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( X != Y )
      | pp(fequal_option_nat(X,Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ~ pp(fequal_option_nat(X,Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fTrue,X,Y) = X ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R),aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y) = X ).

tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R)) ).

tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
    ! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R)) ).

tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
    ! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R),Q) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na))) ).

%------------------------------------------------------------------------------